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Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 738491 10 pageshttpdxdoiorg1011552013738491
Research ArticleGeneralized119867(sdot sdot sdot)-120578-Cocoercive Operators andGeneralized Set-Valued Variational-Like Inclusions
Shamshad Husain1 Sanjeev Gupta1 and Vishnu Narayan Mishra2
1 Department of Applied Mathematics Faculty of Engineering amp Technology Aligarh Muslim University Aligarh 202002 India2Department of Applied Mathematics amp Humanities Sardar Vallabhbhai National Institute of TechnologyIchchhanath Mahadev Road Surat 395 007 India
Correspondence should be addressed to Sanjeev Gupta guptasanmpgmailcom
Received 18 March 2013 Revised 4 May 2013 Accepted 8 May 2013
Academic Editor Kaleem R Kazmi
Copyright copy 2013 Shamshad Husain et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We investigate a new class of cocoercive operators named generalized119867(sdot sdot sdot)-120578-cocoercive operators in Hilbert spaces We provethat generalized 119867(sdot sdot sdot)-120578-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolventoperators associated with119867(sdot sdot)-cocoercive operators to the generalized119867(sdot sdot sdot)-120578-cocoercive operators Some examples are givento justify the definition of generalized 119867(sdot sdot sdot)-120578-cocoercive operators Further we consider a generalized set-valued variational-like inclusion problem involving generalized 119867(sdot sdot sdot)-120578-cocoercive operator In terms of the new resolvent operator techniquewe give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusionsFurthermore we discuss the convergence criteria of iterative algorithm under some suitable conditions Our results can be viewedas a generalization of some known results in the literature
1 Introduction
Variational inclusions as the generalization of variationalinequalities have been widely studied in recent years Oneof the most interesting and important problems in thetheory of variational inclusions include variational quasi-variational and variational-like inequalities as special casesFor applications of variational inclusions see for example[1] Various kinds of iterative methods have been studiedto solve the variational inclusions Among these methodsthe resolvent operator technique for the study of variationalinclusions has been widely used bymany authors For detailswe refer to [2ndash16]
Recently Fang and Huang Kazmi and Khan and Lanet al investigated several resolvent operators for generalizedoperators such as 119867-monotone [5] 119867-accretive [6] (119875 120578)-proximal point [11] (119875 120578)-accretive [12] (119867 120578)-monotone[7] (119860 120578)-accretive [13] and mappings Very recently ZouandHuang [16] introduced and studied119867(sdot sdot)-accretive oper-ators Kazmi et al [8ndash10] introduced and studied generalized
119867(sdot sdot)-accretive operators 119867(sdot sdot)-120578-proximal point map-ping Xu and Wang [15] introduced and studied (119867(sdot sdot) 120578)-monotone operators and Ahmad et al [2] introduced andstudied119867(sdot sdot)-cocoercive operators
Motivated by the recent work going in this direction weconsider a class of cocoercive operators called generalized119867(sdot sdot sdot)-120578-cocoercive a natural generalization of monotone(accretive) operators in Hilbert (Banach) spaces For detailswe refer to [2 5ndash7 13ndash16]We prove that generalized119867(sdot sdot sdot)-120578-cocoercive operator is single-valued and Lipschitz continu-ous and extends the concept of resolvent operators associatedwith 119867(sdot sdot)-cocoercive operators to the generalized 119867(sdot sdot sdot)-120578-cocoercive operators Further we consider the general-ized set-valued variational-like inclusion problem involvinggeneralized119867(sdot sdot sdot)-120578-cocoercive operator in Hilbert spacesUsing new a resolvent operator technique we prove theexistence of solutions and suggest an iterative algorithm forthe generalized set-valued variational-like inclusions Fur-thermore we discuss the convergence criteria of the iterativealgorithm under some suitable conditions Our results can
2 Journal of Mathematics
be viewed as an extension and generalization of some knownresults [2 15 16] For illustration of Definitions 2 and 7 andexamples 23 and 32 are given respectively
2 Preliminaries
Throughout this paper we suppose that 119883 is a real Hilbertspace endowedwith a norm sdot and an inner product ⟨sdot sdot⟩ 2119883(resp CB(119883)) is the family of all the nonempty (resp closedand bounded) subsets of119883 andD(sdot sdot) is theHausdorffmetricon CB(119883) defined by
D (119875 119876) = max sup119909isin119875
119889 (119909 119876) sup119910isin119876
119889 (119875 119910)
forall119875 119876 isin CB (119883)
(1)
where 119889(119909 119876) = inf119910isin119876
119909 minus 119910 and 119889(119875 119910) = inf119909isin119875
119909 minus 119910In the sequel let us recall some concepts
Definition 1 (see [4 17]) Let119875 119883 rarr 119883 and 120578 119883times119883 rarr 119883
be two mappings Then 119875 is said to be
(i) 120578-monotone if
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 0 forall119909 119910 isin 119883 (2)
(ii) 1205751-120578-stronglymonotone if there exists a constant 120575
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205751
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2 forall119909 119910 isin 119883 (3)
(iii) 1205831-120578-cocoercive if there exists a constant 120583
1gt 0 such
that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205831
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817
2
forall119909 119910 isin 119883
(4)
(iv) 1205741-120578-relaxed cocoercive if there exists a constant 120574
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge (minus1205741)1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817
2
forall119909 119910 isin 119883
(5)
(v) 120582119875-Lipschitz continuous if there exists a constant
120582119875gt 0 such that
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817 le 120582
119875
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)
(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119883 then definitions(i) to (iv) reduce to the Definitions of monotonicity strongmonotonicity [18] cocoercivity [19] and relaxed cocoerciverespectively
Definition 2 Let 119875119876 119877 119883 rarr 119883 120578 119883 times 119883 rarr 119883 and119867 119883 times 119883 times 119883 rarr 119883 be the single-valued mappings Then
(i) 119867(119875 sdot sdot) is said to be 120583-120578-cocoercive with respect to 119875if there exists a constant 120583 gt 0 such that
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge 1205831003817100381710038171003817119875119909 minus 119875119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(9)
(ii) 119867(sdot 119876 sdot) is said to be 120574-120578-relaxed cocoercive withrespect to 119876 if there exists a constant 120574 gt 0 such that
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge (minus120574)1003817100381710038171003817119876119909 minus 119876119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(10)
(iii) 119867(sdot sdot 119877) is said to be 120575-120578-strongly monotone withrespect to 119877 if there exists a constant 120575 gt 0 such that
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 1205751003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(11)
(iv) 119867(119875 sdot sdot) is said to be 1199031-Lipschitz continuous with
respect to 119875 if there exists a constant 1199031gt 0 such that
1003817100381710038171003817119867 (119875119909 sdot sdot) minus 119867 (119875119910 sdot sdot)1003817100381710038171003817 le 1199031
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)
(v) 119867(sdot 119876 sdot) is said to be 1199032-Lipschitz continuous with
respect to 119876 if there exists a constant 1199032gt 0 such that
1003817100381710038171003817119867 (sdot 119876119909 sdot) minus 119867 (sdot 119876119910 sdot)1003817100381710038171003817 le 1199032
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (13)
(vi) 119867(sdot sdot 119877) is said to be 1199033-Lipschitz continuous with
respect to 119877 if there exists a constant 1199033gt 0 such that
1003817100381710038171003817119867 (sdot sdot 119877119909) minus 119867 (sdot sdot 119877119910)1003817100381710038171003817 le 1199033
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (14)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119867 and 119867(sdot sdot sdot) =
119867(sdot sdot) then Definitions (i)-(ii) are reduced to the definitionof cocoercivity and relaxed cocoercive [2] respectively and(iii) reduces to strong accretivity [16]
Example 3 Let 119883 = R2 with usual inner product Let 120578
R2 timesR2 rarr R2 119875 119876 119877 R2 rarr R2 be defined by
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
be viewed as an extension and generalization of some knownresults [2 15 16] For illustration of Definitions 2 and 7 andexamples 23 and 32 are given respectively
2 Preliminaries
Throughout this paper we suppose that 119883 is a real Hilbertspace endowedwith a norm sdot and an inner product ⟨sdot sdot⟩ 2119883(resp CB(119883)) is the family of all the nonempty (resp closedand bounded) subsets of119883 andD(sdot sdot) is theHausdorffmetricon CB(119883) defined by
D (119875 119876) = max sup119909isin119875
119889 (119909 119876) sup119910isin119876
119889 (119875 119910)
forall119875 119876 isin CB (119883)
(1)
where 119889(119909 119876) = inf119910isin119876
119909 minus 119910 and 119889(119875 119910) = inf119909isin119875
119909 minus 119910In the sequel let us recall some concepts
Definition 1 (see [4 17]) Let119875 119883 rarr 119883 and 120578 119883times119883 rarr 119883
be two mappings Then 119875 is said to be
(i) 120578-monotone if
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 0 forall119909 119910 isin 119883 (2)
(ii) 1205751-120578-stronglymonotone if there exists a constant 120575
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205751
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2 forall119909 119910 isin 119883 (3)
(iii) 1205831-120578-cocoercive if there exists a constant 120583
1gt 0 such
that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205831
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817
2
forall119909 119910 isin 119883
(4)
(iv) 1205741-120578-relaxed cocoercive if there exists a constant 120574
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge (minus1205741)1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817
2
forall119909 119910 isin 119883
(5)
(v) 120582119875-Lipschitz continuous if there exists a constant
120582119875gt 0 such that
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817 le 120582
119875
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)
(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119883 then definitions(i) to (iv) reduce to the Definitions of monotonicity strongmonotonicity [18] cocoercivity [19] and relaxed cocoerciverespectively
Definition 2 Let 119875119876 119877 119883 rarr 119883 120578 119883 times 119883 rarr 119883 and119867 119883 times 119883 times 119883 rarr 119883 be the single-valued mappings Then
(i) 119867(119875 sdot sdot) is said to be 120583-120578-cocoercive with respect to 119875if there exists a constant 120583 gt 0 such that
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge 1205831003817100381710038171003817119875119909 minus 119875119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(9)
(ii) 119867(sdot 119876 sdot) is said to be 120574-120578-relaxed cocoercive withrespect to 119876 if there exists a constant 120574 gt 0 such that
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge (minus120574)1003817100381710038171003817119876119909 minus 119876119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(10)
(iii) 119867(sdot sdot 119877) is said to be 120575-120578-strongly monotone withrespect to 119877 if there exists a constant 120575 gt 0 such that
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 1205751003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(11)
(iv) 119867(119875 sdot sdot) is said to be 1199031-Lipschitz continuous with
respect to 119875 if there exists a constant 1199031gt 0 such that
1003817100381710038171003817119867 (119875119909 sdot sdot) minus 119867 (119875119910 sdot sdot)1003817100381710038171003817 le 1199031
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)
(v) 119867(sdot 119876 sdot) is said to be 1199032-Lipschitz continuous with
respect to 119876 if there exists a constant 1199032gt 0 such that
1003817100381710038171003817119867 (sdot 119876119909 sdot) minus 119867 (sdot 119876119910 sdot)1003817100381710038171003817 le 1199032
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (13)
(vi) 119867(sdot sdot 119877) is said to be 1199033-Lipschitz continuous with
respect to 119877 if there exists a constant 1199033gt 0 such that
1003817100381710038171003817119867 (sdot sdot 119877119909) minus 119867 (sdot sdot 119877119910)1003817100381710038171003817 le 1199033
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (14)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119867 and 119867(sdot sdot sdot) =
119867(sdot sdot) then Definitions (i)-(ii) are reduced to the definitionof cocoercivity and relaxed cocoercive [2] respectively and(iii) reduces to strong accretivity [16]
Example 3 Let 119883 = R2 with usual inner product Let 120578
R2 timesR2 rarr R2 119875 119876 119877 R2 rarr R2 be defined by
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969