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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 419701 8 pageshttpdxdoiorg1011552013419701
Research ArticleThe Strong Fuzzy Henstock Integrals and Discontinuous FuzzyDifferential Equations
Yabin Shao12 and Huanhuan Zhang1
1 College of Mathematics and Computer Science Northwest University for Nationalities Lanzhou 730030 China2Department of Mathematics Sichuan University Chengdu 610065 China
Correspondence should be addressed to Yabin Shao yb-shao163com
Received 24 April 2013 Accepted 24 September 2013
Academic Editor Mehmet Sezer
Copyright copy 2013 Y Shao and H Zhang 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 generalized the existence theorems and the continuous dependence of a solution on parameters for initial problems of fuzzydiscontinuous differential equation by the strong fuzzy Henstock integral and its controlled convergence theorem
1 Introduction
The Cauchy problems for fuzzy differential equations havebeen studied by several authors [1ndash6] on the metric space(119864
119899 119863) of normal fuzzy convex set with the distance119863 givenby the maximum of the Hausdorff distance between thecorresponding level sets In [4] Nieto proved that the Cauchyproblem has a uniqueness result if 119891 is continuous andbounded In [1 3 7ndash9] the authors presented a uniquenessresult when 119891 satisfies a Lipschitz condition For a generalreference to fuzzy differential equations see a recent bookby Lakshmikantham and Mohapatra [10] and referencestherein In 2002 Xue and Fu [11] established the solutionsto fuzzy differential equations with right-hand side functionssatisfying Caratheodory conditions on a class of Lipschitzfuzzy setsHowever there are discontinuous systems inwhichthe right-hand side functions 119891 [119886 119887] times 119864
119899 rarr 119864119899 arenot integrable in the sense of Kaleva [1] on certain intervalsand their solutions are not absolute continuous functions Toillustrate we consider the following example
Example 1 Consider the following discontinuous system
1199091015840
(119905) = ℎ (119905) 119909 (0) = 119860
119892 (119905) =
2119905 sin 1
1199052minus2
119905cos 1
1199052 119905 = 0
0 119905 = 0
119860 (119904) =
119904 0 le 119904 le 1
2 minus 119904 1 lt 119904 le 2
0 others
ℎ (119905) = 120594|119892(119905)|
+ 119860
(1)
Then ℎ(119905) = 120594|119892(119905)|
+119860 is not integrable in the sense of KalevaHowever the above system has the following solution
119909 (119905) = 120594|119866(119905)|
+ 119860119905 (2)
where
119866 (119905) =
1199052 sin 1
1199052 119905 = 0
0 119905 = 0(3)
It is well known that the Henstock integral is designedto integrate highly oscillatory functions which the Lebesgueintegral fails to do It is known as nonabsolute integrationand is a powerful tool It is well known that the Henstockintegral includes the Riemann improper Riemann Lebesgueand Newton integrals [12 13] Though such an integral wasdefined by Denjoy in 1912 and also by Perron in 1914 itwas difficult to handle using their definitions But withthe Riemann-type definition introduced more recently byHenstock [12] in 1963 and also independently by Kurzweil
2 Journal of Applied Mathematics
[13] the definition is now simple and furthermore the proofinvolving the integral also turns out to be easy For moredetailed results about the Henstock integral we refer to[14] Recently Wu and Gong [15 16] have combined thefuzzy set theory and nonabsolute integration theory andthey discussed the fuzzyHenstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration Inorder to complete the theory of fuzzy calculus and tomeet thesolving need of transferring a fuzzy differential equation intoa fuzzy integral equation Gong and Shao [17 18] defined thestrong fuzzy Henstock integrals and discussed some of theirproperties and the controlled convergence theorem
In this paper according to the idea of [19] and usingthe concept of generalized differentiability [20] we willprove other controlled convergence theorems for the strongfuzzy Henstock integrals which will be of foundationalsignificance for studying the existence and uniqueness ofsolutions to the fuzzy discontinuous systems As we knowwe inevitably use the controlled convergence theorems forsolving the numerical solutions of differential equations Asthe main outcomes we will deal with the Cauchy problem ofdiscontinuous fuzzy systems as follows
1199091015840
(119905) = 119891 (119905 119909)
119909 (120591) = 120585 isin 119864119899
(4)
where 119891 119880 rarr 119864119899 is a strong fuzzy Henstock integrablefunction and
119880 = (119905 119909) |119905 minus 120591| le 119886 119909 isin 119864119899
119863 (119909 120585) le 119887 (5)
To make our analysis possible we will first recall somebasic results of fuzzy numbers and give some definitionsof absolutely continuous fuzzy-number-valued function Inaddition we present the concept of generalized differentia-bility In Section 3 we present the concept of strong fuzzyHenstock integrals and we prove a controlled convergencetheorem for the strong fuzzyHenstock integrals In Section 4we deal with the Cauchy problem of discontinuous fuzzysystems And in Section 5 we present some concludingremarks
2 Preliminaries
Let119875119896(119877119899) denote the family of all nonempty compact convex
subset of119877119899 and define the addition and scalarmultiplicationin 119875
119896(119877119899) as usual Let 119860 and 119861 be two nonempty bounded
subsets of 119877119899 The distance between119860 and 119861 is defined by theHausdorff metric [21] as follows
119889119867(119860 119861) = maxsup
119886isin119860
inf119887isin119861
119886 minus 119887 sup119887isin119861
inf119886isin119860
119887 minus 119886 (6)
Denote that 119864119899 = 119906 119877119899 rarr [0 1] | 119906 satisfies (1)ndash(4)below is a fuzzy number space where
(1) 119906 is normal that is there exists an 1199090isin 119877119899 such that
119906(1199090) = 1
(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge
min119906(119909) 119906(119910) for any 119909 119910 isin 119877119899 and 0 le 120582 le 1(3) 119906 is upper semicontinuous(4) [119906]0 = cl119909 isin 119877119899 | 119906(119909) gt 0 is compact
For 0 lt 120572 le 1 denote that [119906]120572 = 119909 isin 119877119899 | 119906(119909) ge 120572Then from the above (1)ndash(4) it follows that the 120572-level set[119906]
120572
isin 119875119896(119877119899) for all 0 le 120572 lt 1
According to Zadehrsquos extension principle we have addi-tion and scalar multiplication in fuzzy number space 119864119899 asfollows [21]
where 119906 V isin 119864119899 and 0 le 120572 le 1Define119863 119864119899 times 119864119899 rarr [0infin)
119863 (119906 V) = sup 119889119867([119906]
120572
[V]120572) 120572 isin [0 1] (8)
where 119889 is the Hausdorff metric defined in 119875119896(119877119899) Then it is
easy to see that119863 is a metric in 119864119899 Using the results [22] weknow that
(1) (119864119899 119863) is a complete metric space(2) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(3) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V 119908 isin 119864119899 and 120582 isin 119877
Let 119909 119910 isin 119864119899 If there exist 119911 isin 119864119899 such that 119909 = 119910 + 119911then 119911 is called the 119867-difference of 119909 and 119910 and is denotedby 119909 minus
119867
119910 As mentioned above which always is called thecondition (119867) It is well known that the 119867-derivative forfuzzy-number-functions was initially introduced by Puri etal [5 23] and it is based on the condition (119867) of sets Wenote that this definition is fairly strong because the familyof fuzzy-number-valued functions 119867-differentiable is veryrestrictive For example the fuzzy-number-valued function119891 [119886 119887] rarr 119877F defined by 119891(119909) = 119862 sdot 119892(119909) where 119862 isa fuzzy number sdot is the scalar multiplication (in the fuzzycontext) and 119892 [119886 119887] rarr 119877
+ with 1198921015840(1199050) lt 0 is not 119867-
differentiable in 1199050(see [20 24]) To avoid the above difficulty
in this paper we consider a more general definition of aderivative for fuzzy-number-valued functions enlarging theclass of differentiable fuzzy-number-valued functions whichhas been introduced in [20]
Definition 2 (see [20]) Let 119891 (119886 119887) rarr 119864119899 and 119909
0isin
(119886 119887) One says that 119891 is differentiable at 1199090 if there exists
an element 1198911015840(1199050) isin 119864119899 such that
(1) for all ℎ gt 0 sufficiently small there exists 119891(1199090+
ℎ) minus119867119891(119909
0) 119891(119909
0) minus
119867119891(119909
0minus ℎ) and the limits (in
the metric119863)
limℎrarr0
119891 (1199090+ ℎ) minus
119867119891 (119909
0)
ℎ= lim
ℎrarr0
119891 (1199090) minus
119867119891 (119909
0minus ℎ)
ℎ
= 1198911015840
(1199090)
(9)
or
Journal of Applied Mathematics 3
(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(10)
or
(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus
119867119891(119909
0) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090+ ℎ) minus
119867119891 (119909
0)
ℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(11)
or
(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0) minus
119867119891(119909
0minus ℎ) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090) minus
119867119891 (119909
0minus ℎ)
ℎ
= 1198911015840
(1199090)
(12)
(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)
3 The Convergence Theorem of Strong FuzzyHenstock Integral
In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions
Definition 3 (see [18]) A fuzzy-number-valued function 119891
is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909
1le 119909
2le sdot sdot sdot le 119909
119899le 119887 one has 119891([119909
119894 119909
119895]) (1 le
119894 lt 119895 le 119899) that exists and 119891([119909119894 119909
119895]) = sum
119895minus1
119896=119894119891([119909
119896 119909
119896+1]) or
119891([119909119895 119909
119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909
119895 119909
119894]) =
(minus1) sdot sum119895minus1
119896=119894119891([119909
119896+1 119909
119896]) For convenience 119891([119904 119905]) denotes
119891(119905) minus119867119891(119904)
Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865
on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has
sum119894isin119870119899
119863(119891 (120585119894) (V
119894minus 119906
119894) 119865 ([119906
119894 V
119894]))
+ sum119895isin119868119899
119863(119891 (120585119895) (V
119895minus 119906
119895) (minus1) sdot 119865 ([119906
119895 V
119895minus1])) lt 120576
(13)
where 119870119899= 119894 isin 1 2 119899 such that 119865([119909
119894minus1 119909
119894]) is a fuzzy
number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909
119895 119909
119895minus1]) is a
fuzzy number One writes 119891 isin SFH[119886 119887]
Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast
120575(119883) if for every 120576 gt 0 there exists
120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883
119894satisfyingsum119899
119894=1|V minus 119906| lt 120578 one has
sum119863(119865[119906 V]) lt 120576
Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed
sets 119883119894 such that on each119883
119894 119865 is 119860119862lowast
120575(119883
119894)
Definition 7 The sequence of fuzzy-number-function 119865119899 is
119880119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883
119894such
that 119865119899 is 119880119860119862lowast
120575for each 119894 independent of 119899
Definition 8 Let 119865119899 be a sequence of fuzzy-number-
function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable
(i) The sequence of fuzzy-number-function 119865119899 is P-
Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and
if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865
119898(119875) 119865
119899(119875)) lt 120576 for
all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865
119899 is gen-
eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865
119899 isP-Cauchy
Theorem 9 Let the following conditions be satisfied
(i) 119891119899119883(119909) rarr 119891
119909ae on [119886 119887] as 119899 rarr infin where each
119891119899119883
is strong Fuzzy Henstock integrable on [119886 119887]
(ii) the primitives 119865119899119883
of 119891119899119883
are 119880119860119862lowast
120575with closed set119883
in [119886 119887]
Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with
the primitive 119865119883(119909)
Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865
119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem
[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891
119899(120585) 119891
119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider
the following in which 119875 is a 120575-fine division of [119909 119910] and
4 Journal of Applied Mathematics
119875 = 1198751cup119875
2so that119875
1contains the intervalswith the associated
points 120585 notin 119866 and 1198752otherwise
119863(119865119899119883
(119909 119910) 119865119898119883
(119909 119910))
= (1198752)sum119863(119865
119899119883(119906 V) 119865
119898119883(119906 V))
le sum119863(119865119899119883
(119906 V) 119891119899119883
(120585) (V minus 119906))
+sum119863(119865119898119883
(119906 V) 119891119898119883
(120585) (V minus 119906))
+sum119863(119891119898119883
(120585) (V minus 119906) 119891119898119883
(120585) (V minus 119906))
+sum119863(119865119899119883
(V) 119865119899119883
(119906)) +sum119863(119865119898119883
(V) 119865119898119883
(119906))
lt 120576 (4 + 119887 minus 119886)
(14)
Hence for any 120575-fine partial division 119875 of [119886 119887] we have
for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-
eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883
is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883
Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued
function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that
(1) sup119899119863(sum
119869119896isin1198751
119865119899(119869119896) sum
119871ℎisin1198752
119865119899(119871
ℎ)) lt 120576 for each 119875
1
1198752isin prod(119883 120575)
(2) with |(cup1198751)Δ(cap119875
2)| lt 120578
(b) A sequence 119865119899 of fuzzy-number-valued function is
uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883
119894are
measurable sets and 119865 is uniformly 119860119862nabla on each119883119894
Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast
120575
Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla
on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0
such that (1) holds in Definition 10 for each 1198751 119875
2isin prod(119883 120575)
satisfying condition (2) We take 119875 = ([119888119896 119889
119896] 119909
119896)119901
119894=1with
sum119896|119889
119896minus 119888
119896| lt 120578 and put 119875
1= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119862
119896 119889
119896ge
0) and 1198752= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119888119896 119889
119896lt 0) So we have
|(cup1198751)Δ(cap119875
2)| = sum
119901
119896=1|119889
119896minus 119888
119896| lt 120578 Then by condition (1)
in Definition 10 we have
sup119899
119901
sum119896=1
119863(119865119899(119888119896) 119865
119899(119889
119896))
= sup119863( sum
(119888119896 119889119896)isin1198751
119865119899(119888119896 119889
119896)
sum(119888119896119889119896)isin1198752
119865119899(119888119896 119889
119896)) lt 120576
(16)
Hence we have that 119865 is uniformly 119860119862119866lowast
120575
We get the following theorem byTheorems 9 and 11
Theorem 12 Let the following conditions be satisfied
(i) 119891119899119883
rarr 119891119883ae in [119886 119887]where each119891
119899119883is strong fuzzy
Henstock integrable on [119886 119887](ii) the primitives 119865
119899119883of 119891
119899119883are119880119860119862nabla(119883)with closed set
119883 in [119886 119887]
Then 119891119883
is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865
119883
Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866
120575for a fuzzy-number-valued function
Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862
120575on 119883 if for each
120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873
119894=1119863(119865(119888
119894) 119865(119889
119894)) lt 120576 forsum119873
119894=1(119889
119894minus119888
119894) lt 120578 A fuzzy-number-
valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a
sequence of set 119883119894 such that the function 119865 is 119860119862
120575(119883
119894) for
each 119894
Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0
there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886
119894 119887119894] satisfying sum119899
119894=1|119887119894minus 119886
119894| lt 120578
where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886
119894 119887119894]) lt 120576 where 120596
denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886
119894 119887119894]) =
sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued
function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883
119894 such that on each 119883
119894 119865 is
119860119862lowast(119883119894)
Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if
and only if it is 119860119862119866lowast on [119886 119887]
Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied
(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin
where each 119891119899is strong fuzzy Henstock integrable on
[119886 119887]
Journal of Applied Mathematics 5
(2) the primitives 119865119899(119909) = (SFH) int119909
119886
119891119899(119904)d119909 of 119891
119899are
119880119860119862119866120575uniformly in 119899
(3) the sequence 119865119899(119909) converges uniformly to a con-
tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (17)
If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]
where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable
Proof By condition (2) there exists a sequence 119883119894 such that
119865119899isin 119880119860119862
120575in 119883
119894a bounded closed set with bounds 119886 and
119887 and put 119883 = 119883119894 We note that 119865
119899(119909) rarr 119865(119909) we have
119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866
120575on [119886 119887] ByTheorem 15
119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin
119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact
let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866
119899linearly
to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866
119899and119866 are119880119860119862 on [119886 119887] By condition
(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888
119896 119889
119896] be the intervals
contiguous to 119883 Then we have 119863(1198661015840
119899) le 119872
119896 We define a
fuzzy-number-valued function as follows
1198661015840
(119909) =119866119899(119889
119896) minus
119867119866119899(119888119896)
119889119896minus 119888
119896
119909 isin (119888119896 119889
119896) (18)
Consequently 1198661015840
119899(119909) converges on (119888
119896 119889
119896) Hence 1198661015840
119899con-
verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840
119899(119909) =
119891119899
rarr 119891 on 119883 Therefore we have 1198661015840
119899(119909) = 119892(119909) =
119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]
by Theorem 15 Therefore there exists an 119860119862119866120575function on
[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (19)
4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations
In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation
Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909
Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880
Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le
119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896
119899(119905) of step functions defined on |119905 minus 120591| le
119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)
almost everywhere as 119899 rarr infin Let 119865119899(119905) = int
119905
120591
119891(119904 119896119899(119904))d119904
Then 119865119899(119905) is 119880119860119862119866
120575uniformly in 119899 and equicontinuous
By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable
Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr
119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions
(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868
(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868
Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)
Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585
Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus
By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let
120601 (119905)=120595 (119905)+int119905
120591
119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
[13] the definition is now simple and furthermore the proofinvolving the integral also turns out to be easy For moredetailed results about the Henstock integral we refer to[14] Recently Wu and Gong [15 16] have combined thefuzzy set theory and nonabsolute integration theory andthey discussed the fuzzyHenstock integrals of fuzzy-number-valued functions which extended Kaleva [1] integration Inorder to complete the theory of fuzzy calculus and tomeet thesolving need of transferring a fuzzy differential equation intoa fuzzy integral equation Gong and Shao [17 18] defined thestrong fuzzy Henstock integrals and discussed some of theirproperties and the controlled convergence theorem
In this paper according to the idea of [19] and usingthe concept of generalized differentiability [20] we willprove other controlled convergence theorems for the strongfuzzy Henstock integrals which will be of foundationalsignificance for studying the existence and uniqueness ofsolutions to the fuzzy discontinuous systems As we knowwe inevitably use the controlled convergence theorems forsolving the numerical solutions of differential equations Asthe main outcomes we will deal with the Cauchy problem ofdiscontinuous fuzzy systems as follows
1199091015840
(119905) = 119891 (119905 119909)
119909 (120591) = 120585 isin 119864119899
(4)
where 119891 119880 rarr 119864119899 is a strong fuzzy Henstock integrablefunction and
119880 = (119905 119909) |119905 minus 120591| le 119886 119909 isin 119864119899
119863 (119909 120585) le 119887 (5)
To make our analysis possible we will first recall somebasic results of fuzzy numbers and give some definitionsof absolutely continuous fuzzy-number-valued function Inaddition we present the concept of generalized differentia-bility In Section 3 we present the concept of strong fuzzyHenstock integrals and we prove a controlled convergencetheorem for the strong fuzzyHenstock integrals In Section 4we deal with the Cauchy problem of discontinuous fuzzysystems And in Section 5 we present some concludingremarks
2 Preliminaries
Let119875119896(119877119899) denote the family of all nonempty compact convex
subset of119877119899 and define the addition and scalarmultiplicationin 119875
119896(119877119899) as usual Let 119860 and 119861 be two nonempty bounded
subsets of 119877119899 The distance between119860 and 119861 is defined by theHausdorff metric [21] as follows
119889119867(119860 119861) = maxsup
119886isin119860
inf119887isin119861
119886 minus 119887 sup119887isin119861
inf119886isin119860
119887 minus 119886 (6)
Denote that 119864119899 = 119906 119877119899 rarr [0 1] | 119906 satisfies (1)ndash(4)below is a fuzzy number space where
(1) 119906 is normal that is there exists an 1199090isin 119877119899 such that
119906(1199090) = 1
(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge
min119906(119909) 119906(119910) for any 119909 119910 isin 119877119899 and 0 le 120582 le 1(3) 119906 is upper semicontinuous(4) [119906]0 = cl119909 isin 119877119899 | 119906(119909) gt 0 is compact
For 0 lt 120572 le 1 denote that [119906]120572 = 119909 isin 119877119899 | 119906(119909) ge 120572Then from the above (1)ndash(4) it follows that the 120572-level set[119906]
120572
isin 119875119896(119877119899) for all 0 le 120572 lt 1
According to Zadehrsquos extension principle we have addi-tion and scalar multiplication in fuzzy number space 119864119899 asfollows [21]
where 119906 V isin 119864119899 and 0 le 120572 le 1Define119863 119864119899 times 119864119899 rarr [0infin)
119863 (119906 V) = sup 119889119867([119906]
120572
[V]120572) 120572 isin [0 1] (8)
where 119889 is the Hausdorff metric defined in 119875119896(119877119899) Then it is
easy to see that119863 is a metric in 119864119899 Using the results [22] weknow that
(1) (119864119899 119863) is a complete metric space(2) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(3) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V 119908 isin 119864119899 and 120582 isin 119877
Let 119909 119910 isin 119864119899 If there exist 119911 isin 119864119899 such that 119909 = 119910 + 119911then 119911 is called the 119867-difference of 119909 and 119910 and is denotedby 119909 minus
119867
119910 As mentioned above which always is called thecondition (119867) It is well known that the 119867-derivative forfuzzy-number-functions was initially introduced by Puri etal [5 23] and it is based on the condition (119867) of sets Wenote that this definition is fairly strong because the familyof fuzzy-number-valued functions 119867-differentiable is veryrestrictive For example the fuzzy-number-valued function119891 [119886 119887] rarr 119877F defined by 119891(119909) = 119862 sdot 119892(119909) where 119862 isa fuzzy number sdot is the scalar multiplication (in the fuzzycontext) and 119892 [119886 119887] rarr 119877
+ with 1198921015840(1199050) lt 0 is not 119867-
differentiable in 1199050(see [20 24]) To avoid the above difficulty
in this paper we consider a more general definition of aderivative for fuzzy-number-valued functions enlarging theclass of differentiable fuzzy-number-valued functions whichhas been introduced in [20]
Definition 2 (see [20]) Let 119891 (119886 119887) rarr 119864119899 and 119909
0isin
(119886 119887) One says that 119891 is differentiable at 1199090 if there exists
an element 1198911015840(1199050) isin 119864119899 such that
(1) for all ℎ gt 0 sufficiently small there exists 119891(1199090+
ℎ) minus119867119891(119909
0) 119891(119909
0) minus
119867119891(119909
0minus ℎ) and the limits (in
the metric119863)
limℎrarr0
119891 (1199090+ ℎ) minus
119867119891 (119909
0)
ℎ= lim
ℎrarr0
119891 (1199090) minus
119867119891 (119909
0minus ℎ)
ℎ
= 1198911015840
(1199090)
(9)
or
Journal of Applied Mathematics 3
(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(10)
or
(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus
119867119891(119909
0) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090+ ℎ) minus
119867119891 (119909
0)
ℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(11)
or
(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0) minus
119867119891(119909
0minus ℎ) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090) minus
119867119891 (119909
0minus ℎ)
ℎ
= 1198911015840
(1199090)
(12)
(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)
3 The Convergence Theorem of Strong FuzzyHenstock Integral
In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions
Definition 3 (see [18]) A fuzzy-number-valued function 119891
is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909
1le 119909
2le sdot sdot sdot le 119909
119899le 119887 one has 119891([119909
119894 119909
119895]) (1 le
119894 lt 119895 le 119899) that exists and 119891([119909119894 119909
119895]) = sum
119895minus1
119896=119894119891([119909
119896 119909
119896+1]) or
119891([119909119895 119909
119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909
119895 119909
119894]) =
(minus1) sdot sum119895minus1
119896=119894119891([119909
119896+1 119909
119896]) For convenience 119891([119904 119905]) denotes
119891(119905) minus119867119891(119904)
Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865
on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has
sum119894isin119870119899
119863(119891 (120585119894) (V
119894minus 119906
119894) 119865 ([119906
119894 V
119894]))
+ sum119895isin119868119899
119863(119891 (120585119895) (V
119895minus 119906
119895) (minus1) sdot 119865 ([119906
119895 V
119895minus1])) lt 120576
(13)
where 119870119899= 119894 isin 1 2 119899 such that 119865([119909
119894minus1 119909
119894]) is a fuzzy
number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909
119895 119909
119895minus1]) is a
fuzzy number One writes 119891 isin SFH[119886 119887]
Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast
120575(119883) if for every 120576 gt 0 there exists
120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883
119894satisfyingsum119899
119894=1|V minus 119906| lt 120578 one has
sum119863(119865[119906 V]) lt 120576
Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed
sets 119883119894 such that on each119883
119894 119865 is 119860119862lowast
120575(119883
119894)
Definition 7 The sequence of fuzzy-number-function 119865119899 is
119880119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883
119894such
that 119865119899 is 119880119860119862lowast
120575for each 119894 independent of 119899
Definition 8 Let 119865119899 be a sequence of fuzzy-number-
function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable
(i) The sequence of fuzzy-number-function 119865119899 is P-
Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and
if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865
119898(119875) 119865
119899(119875)) lt 120576 for
all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865
119899 is gen-
eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865
119899 isP-Cauchy
Theorem 9 Let the following conditions be satisfied
(i) 119891119899119883(119909) rarr 119891
119909ae on [119886 119887] as 119899 rarr infin where each
119891119899119883
is strong Fuzzy Henstock integrable on [119886 119887]
(ii) the primitives 119865119899119883
of 119891119899119883
are 119880119860119862lowast
120575with closed set119883
in [119886 119887]
Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with
the primitive 119865119883(119909)
Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865
119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem
[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891
119899(120585) 119891
119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider
the following in which 119875 is a 120575-fine division of [119909 119910] and
4 Journal of Applied Mathematics
119875 = 1198751cup119875
2so that119875
1contains the intervalswith the associated
points 120585 notin 119866 and 1198752otherwise
119863(119865119899119883
(119909 119910) 119865119898119883
(119909 119910))
= (1198752)sum119863(119865
119899119883(119906 V) 119865
119898119883(119906 V))
le sum119863(119865119899119883
(119906 V) 119891119899119883
(120585) (V minus 119906))
+sum119863(119865119898119883
(119906 V) 119891119898119883
(120585) (V minus 119906))
+sum119863(119891119898119883
(120585) (V minus 119906) 119891119898119883
(120585) (V minus 119906))
+sum119863(119865119899119883
(V) 119865119899119883
(119906)) +sum119863(119865119898119883
(V) 119865119898119883
(119906))
lt 120576 (4 + 119887 minus 119886)
(14)
Hence for any 120575-fine partial division 119875 of [119886 119887] we have
for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-
eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883
is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883
Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued
function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that
(1) sup119899119863(sum
119869119896isin1198751
119865119899(119869119896) sum
119871ℎisin1198752
119865119899(119871
ℎ)) lt 120576 for each 119875
1
1198752isin prod(119883 120575)
(2) with |(cup1198751)Δ(cap119875
2)| lt 120578
(b) A sequence 119865119899 of fuzzy-number-valued function is
uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883
119894are
measurable sets and 119865 is uniformly 119860119862nabla on each119883119894
Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast
120575
Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla
on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0
such that (1) holds in Definition 10 for each 1198751 119875
2isin prod(119883 120575)
satisfying condition (2) We take 119875 = ([119888119896 119889
119896] 119909
119896)119901
119894=1with
sum119896|119889
119896minus 119888
119896| lt 120578 and put 119875
1= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119862
119896 119889
119896ge
0) and 1198752= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119888119896 119889
119896lt 0) So we have
|(cup1198751)Δ(cap119875
2)| = sum
119901
119896=1|119889
119896minus 119888
119896| lt 120578 Then by condition (1)
in Definition 10 we have
sup119899
119901
sum119896=1
119863(119865119899(119888119896) 119865
119899(119889
119896))
= sup119863( sum
(119888119896 119889119896)isin1198751
119865119899(119888119896 119889
119896)
sum(119888119896119889119896)isin1198752
119865119899(119888119896 119889
119896)) lt 120576
(16)
Hence we have that 119865 is uniformly 119860119862119866lowast
120575
We get the following theorem byTheorems 9 and 11
Theorem 12 Let the following conditions be satisfied
(i) 119891119899119883
rarr 119891119883ae in [119886 119887]where each119891
119899119883is strong fuzzy
Henstock integrable on [119886 119887](ii) the primitives 119865
119899119883of 119891
119899119883are119880119860119862nabla(119883)with closed set
119883 in [119886 119887]
Then 119891119883
is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865
119883
Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866
120575for a fuzzy-number-valued function
Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862
120575on 119883 if for each
120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873
119894=1119863(119865(119888
119894) 119865(119889
119894)) lt 120576 forsum119873
119894=1(119889
119894minus119888
119894) lt 120578 A fuzzy-number-
valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a
sequence of set 119883119894 such that the function 119865 is 119860119862
120575(119883
119894) for
each 119894
Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0
there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886
119894 119887119894] satisfying sum119899
119894=1|119887119894minus 119886
119894| lt 120578
where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886
119894 119887119894]) lt 120576 where 120596
denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886
119894 119887119894]) =
sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued
function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883
119894 such that on each 119883
119894 119865 is
119860119862lowast(119883119894)
Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if
and only if it is 119860119862119866lowast on [119886 119887]
Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied
(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin
where each 119891119899is strong fuzzy Henstock integrable on
[119886 119887]
Journal of Applied Mathematics 5
(2) the primitives 119865119899(119909) = (SFH) int119909
119886
119891119899(119904)d119909 of 119891
119899are
119880119860119862119866120575uniformly in 119899
(3) the sequence 119865119899(119909) converges uniformly to a con-
tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (17)
If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]
where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable
Proof By condition (2) there exists a sequence 119883119894 such that
119865119899isin 119880119860119862
120575in 119883
119894a bounded closed set with bounds 119886 and
119887 and put 119883 = 119883119894 We note that 119865
119899(119909) rarr 119865(119909) we have
119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866
120575on [119886 119887] ByTheorem 15
119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin
119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact
let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866
119899linearly
to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866
119899and119866 are119880119860119862 on [119886 119887] By condition
(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888
119896 119889
119896] be the intervals
contiguous to 119883 Then we have 119863(1198661015840
119899) le 119872
119896 We define a
fuzzy-number-valued function as follows
1198661015840
(119909) =119866119899(119889
119896) minus
119867119866119899(119888119896)
119889119896minus 119888
119896
119909 isin (119888119896 119889
119896) (18)
Consequently 1198661015840
119899(119909) converges on (119888
119896 119889
119896) Hence 1198661015840
119899con-
verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840
119899(119909) =
119891119899
rarr 119891 on 119883 Therefore we have 1198661015840
119899(119909) = 119892(119909) =
119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]
by Theorem 15 Therefore there exists an 119860119862119866120575function on
[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (19)
4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations
In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation
Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909
Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880
Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le
119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896
119899(119905) of step functions defined on |119905 minus 120591| le
119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)
almost everywhere as 119899 rarr infin Let 119865119899(119905) = int
119905
120591
119891(119904 119896119899(119904))d119904
Then 119865119899(119905) is 119880119860119862119866
120575uniformly in 119899 and equicontinuous
By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable
Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr
119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions
(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868
(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868
Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)
Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585
Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus
By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let
120601 (119905)=120595 (119905)+int119905
120591
119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
(2) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(10)
or
(3) for all ℎ gt 0 sufficiently small there exists 119891(1199090+ℎ) minus
119867119891(119909
0) 119891(119909
0minus ℎ) minus
119867119891(119909
0) and the limits
limℎrarr0
119891 (1199090+ ℎ) minus
119867119891 (119909
0)
ℎ= lim
ℎrarr0
119891 (1199090minus ℎ) minus
119867119891 (119909
0)
minusℎ
= 1198911015840
(1199090)
(11)
or
(4) for all ℎ gt 0 sufficiently small there exists 119891(1199090) minus
119867119891(119909
0+ ℎ) 119891(119909
0) minus
119867119891(119909
0minus ℎ) and the limits
limℎrarr0
119891 (1199090) minus
119867119891 (119909
0+ ℎ)
minusℎ= lim
ℎrarr0
119891 (1199090) minus
119867119891 (119909
0minus ℎ)
ℎ
= 1198911015840
(1199090)
(12)
(ℎ and minusℎ at denominators mean (1ℎ)sdot and minus(1ℎ)sdotresp)
3 The Convergence Theorem of Strong FuzzyHenstock Integral
In this section we define the strong Henstock integrals offuzzy-number-valued functions in fuzzy number space 119864119899and we give some properties and controlled convergencetheorem of this integral by using new conditions
Definition 3 (see [18]) A fuzzy-number-valued function 119891
is said to be termed additive on [119886 119887] if for any division119879 119886 le 119909
1le 119909
2le sdot sdot sdot le 119909
119899le 119887 one has 119891([119909
119894 119909
119895]) (1 le
119894 lt 119895 le 119899) that exists and 119891([119909119894 119909
119895]) = sum
119895minus1
119896=119894119891([119909
119896 119909
119896+1]) or
119891([119909119895 119909
119894])(1 le 119894 lt 119895 le 119899) that exists and (minus1) sdot 119891([119909
119895 119909
119894]) =
(minus1) sdot sum119895minus1
119896=119894119891([119909
119896+1 119909
119896]) For convenience 119891([119904 119905]) denotes
119891(119905) minus119867119891(119904)
Definition 4 (see [17 18]) A fuzzy-number-valued function119891 is said to be strong Henstock integrable on [119886 119887] if thereexists a piecewise additive fuzzy-number-valued function 119865
on [119886 119887] such that for every 120576 gt 0 there is a function 120575(120585) gt 0and for any 120575-fine division 119875 = ([119906 V] 120585) of [119886 119887] one has
sum119894isin119870119899
119863(119891 (120585119894) (V
119894minus 119906
119894) 119865 ([119906
119894 V
119894]))
+ sum119895isin119868119899
119863(119891 (120585119895) (V
119895minus 119906
119895) (minus1) sdot 119865 ([119906
119895 V
119895minus1])) lt 120576
(13)
where 119870119899= 119894 isin 1 2 119899 such that 119865([119909
119894minus1 119909
119894]) is a fuzzy
number and 119868119899= 119895 isin 1 2 119899 such that 119865([119909
119895 119909
119895minus1]) is a
fuzzy number One writes 119891 isin SFH[119886 119887]
Definition 5 A fuzzy-number-valued function 119865 defined on119883 sub [119886 119887] is said to be 119860119862lowast
120575(119883) if for every 120576 gt 0 there exists
120578 gt 0 and 120575(120585) gt 0 such that for any 120575-fine partial division119875 = ([119906 V] 120585) with 120585 isin 119883
119894satisfyingsum119899
119894=1|V minus 119906| lt 120578 one has
sum119863(119865[119906 V]) lt 120576
Definition 6 A fuzzy-number-valued function 119865 is said to be119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the union of a sequence of closed
sets 119883119894 such that on each119883
119894 119865 is 119860119862lowast
120575(119883
119894)
Definition 7 The sequence of fuzzy-number-function 119865119899 is
119880119860119862119866lowast
120575on119883 sub [119886 119887] if119883 is the sequence of subsets119883
119894such
that 119865119899 is 119880119860119862lowast
120575for each 119894 independent of 119899
Definition 8 Let 119865119899 be a sequence of fuzzy-number-
function defined on [119886 119887] and let 119909 sub [119886 119887] be measurable
(i) The sequence of fuzzy-number-function 119865119899 is P-
Cauchy on 119864119899 if 119865119899 converges pointwise on 119883 and
if for each 120576 gt 0 there exist 120575(120585) gt 0 on 119883 and apositive integer 119873 such that 119863(119865
119898(119875) 119865
119899(119875)) lt 120576 for
all119898 119899 ge 119873 whenever 119875 is119883-subordinate to 120575(120585)(ii) The sequence of fuzzy-number-function 119865
119899 is gen-
eralized P-Cauchy on 119883 if 119883 can be written as acountable union of measurable sets on each of which119865
119899 isP-Cauchy
Theorem 9 Let the following conditions be satisfied
(i) 119891119899119883(119909) rarr 119891
119909ae on [119886 119887] as 119899 rarr infin where each
119891119899119883
is strong Fuzzy Henstock integrable on [119886 119887]
(ii) the primitives 119865119899119883
of 119891119899119883
are 119880119860119862lowast
120575with closed set119883
in [119886 119887]
Then 119891119883(119909) is strong fuzzy Henstock integrable on [119886 119887] with
the primitive 119865119883(119909)
Proof By (ii) for every 120576 gt 0 there exist a 120575(120585) gt 0 and 120578 gt 0such that for any 120575-fine partial division119875of119883 satisfyingsum |Vminus119906| lt 120578 we have sum119863(119865
119899119883(V 119906) 0) lt 120576 By Egoroff rsquos theorem
[18 Theorem 34] there is an open set 119866 with |119866| lt 120578 suchthat 119863(119891
119899(120585) 119891
119898(120585)) lt 120576 for 119899119898 ge 119873 and 120585 notin 119866 Consider
the following in which 119875 is a 120575-fine division of [119909 119910] and
4 Journal of Applied Mathematics
119875 = 1198751cup119875
2so that119875
1contains the intervalswith the associated
points 120585 notin 119866 and 1198752otherwise
119863(119865119899119883
(119909 119910) 119865119898119883
(119909 119910))
= (1198752)sum119863(119865
119899119883(119906 V) 119865
119898119883(119906 V))
le sum119863(119865119899119883
(119906 V) 119891119899119883
(120585) (V minus 119906))
+sum119863(119865119898119883
(119906 V) 119891119898119883
(120585) (V minus 119906))
+sum119863(119891119898119883
(120585) (V minus 119906) 119891119898119883
(120585) (V minus 119906))
+sum119863(119865119899119883
(V) 119865119899119883
(119906)) +sum119863(119865119898119883
(V) 119865119898119883
(119906))
lt 120576 (4 + 119887 minus 119886)
(14)
Hence for any 120575-fine partial division 119875 of [119886 119887] we have
for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-
eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883
is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883
Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued
function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that
(1) sup119899119863(sum
119869119896isin1198751
119865119899(119869119896) sum
119871ℎisin1198752
119865119899(119871
ℎ)) lt 120576 for each 119875
1
1198752isin prod(119883 120575)
(2) with |(cup1198751)Δ(cap119875
2)| lt 120578
(b) A sequence 119865119899 of fuzzy-number-valued function is
uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883
119894are
measurable sets and 119865 is uniformly 119860119862nabla on each119883119894
Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast
120575
Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla
on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0
such that (1) holds in Definition 10 for each 1198751 119875
2isin prod(119883 120575)
satisfying condition (2) We take 119875 = ([119888119896 119889
119896] 119909
119896)119901
119894=1with
sum119896|119889
119896minus 119888
119896| lt 120578 and put 119875
1= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119862
119896 119889
119896ge
0) and 1198752= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119888119896 119889
119896lt 0) So we have
|(cup1198751)Δ(cap119875
2)| = sum
119901
119896=1|119889
119896minus 119888
119896| lt 120578 Then by condition (1)
in Definition 10 we have
sup119899
119901
sum119896=1
119863(119865119899(119888119896) 119865
119899(119889
119896))
= sup119863( sum
(119888119896 119889119896)isin1198751
119865119899(119888119896 119889
119896)
sum(119888119896119889119896)isin1198752
119865119899(119888119896 119889
119896)) lt 120576
(16)
Hence we have that 119865 is uniformly 119860119862119866lowast
120575
We get the following theorem byTheorems 9 and 11
Theorem 12 Let the following conditions be satisfied
(i) 119891119899119883
rarr 119891119883ae in [119886 119887]where each119891
119899119883is strong fuzzy
Henstock integrable on [119886 119887](ii) the primitives 119865
119899119883of 119891
119899119883are119880119860119862nabla(119883)with closed set
119883 in [119886 119887]
Then 119891119883
is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865
119883
Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866
120575for a fuzzy-number-valued function
Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862
120575on 119883 if for each
120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873
119894=1119863(119865(119888
119894) 119865(119889
119894)) lt 120576 forsum119873
119894=1(119889
119894minus119888
119894) lt 120578 A fuzzy-number-
valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a
sequence of set 119883119894 such that the function 119865 is 119860119862
120575(119883
119894) for
each 119894
Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0
there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886
119894 119887119894] satisfying sum119899
119894=1|119887119894minus 119886
119894| lt 120578
where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886
119894 119887119894]) lt 120576 where 120596
denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886
119894 119887119894]) =
sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued
function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883
119894 such that on each 119883
119894 119865 is
119860119862lowast(119883119894)
Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if
and only if it is 119860119862119866lowast on [119886 119887]
Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied
(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin
where each 119891119899is strong fuzzy Henstock integrable on
[119886 119887]
Journal of Applied Mathematics 5
(2) the primitives 119865119899(119909) = (SFH) int119909
119886
119891119899(119904)d119909 of 119891
119899are
119880119860119862119866120575uniformly in 119899
(3) the sequence 119865119899(119909) converges uniformly to a con-
tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (17)
If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]
where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable
Proof By condition (2) there exists a sequence 119883119894 such that
119865119899isin 119880119860119862
120575in 119883
119894a bounded closed set with bounds 119886 and
119887 and put 119883 = 119883119894 We note that 119865
119899(119909) rarr 119865(119909) we have
119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866
120575on [119886 119887] ByTheorem 15
119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin
119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact
let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866
119899linearly
to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866
119899and119866 are119880119860119862 on [119886 119887] By condition
(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888
119896 119889
119896] be the intervals
contiguous to 119883 Then we have 119863(1198661015840
119899) le 119872
119896 We define a
fuzzy-number-valued function as follows
1198661015840
(119909) =119866119899(119889
119896) minus
119867119866119899(119888119896)
119889119896minus 119888
119896
119909 isin (119888119896 119889
119896) (18)
Consequently 1198661015840
119899(119909) converges on (119888
119896 119889
119896) Hence 1198661015840
119899con-
verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840
119899(119909) =
119891119899
rarr 119891 on 119883 Therefore we have 1198661015840
119899(119909) = 119892(119909) =
119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]
by Theorem 15 Therefore there exists an 119860119862119866120575function on
[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (19)
4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations
In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation
Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909
Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880
Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le
119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896
119899(119905) of step functions defined on |119905 minus 120591| le
119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)
almost everywhere as 119899 rarr infin Let 119865119899(119905) = int
119905
120591
119891(119904 119896119899(119904))d119904
Then 119865119899(119905) is 119880119860119862119866
120575uniformly in 119899 and equicontinuous
By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable
Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr
119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions
(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868
(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868
Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)
Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585
Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus
By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let
120601 (119905)=120595 (119905)+int119905
120591
119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
for 119898 119899 ge 119873 Therefore the fuzzy sequence 119865119899119883 is gen-
eralized P-Cauchy on [119886 119887] Then by (i) we have that 119891119883
is strong fuzzy Henstock integrable on [119886 119887] with primitive119865119883
Definition 10 (a) A sequence 119865119899 of fuzzy-number-valued
function is uniformly119860119862nabla on119883whenever to each 120576 gt 0 thereexist 120578 gt 0 and 120575(119909) gt 0 such that
(1) sup119899119863(sum
119869119896isin1198751
119865119899(119869119896) sum
119871ℎisin1198752
119865119899(119871
ℎ)) lt 120576 for each 119875
1
1198752isin prod(119883 120575)
(2) with |(cup1198751)Δ(cap119875
2)| lt 120578
(b) A sequence 119865119899 of fuzzy-number-valued function is
uniformly 119860119862119866nabla on [119886 119887] if [119886 119887] = cup119894119883119894 where 119883
119894are
measurable sets and 119865 is uniformly 119860119862nabla on each119883119894
Theorem 11 If 119865 is uniformly 119860119862119866nabla then 119865 is uniformly119860119862119866lowast
120575
Proof Let [119886 119887] = cup119894119883119894be such that 119865 is uniformly 119860119862119866nabla
on each 119883119894 So for each 120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0
such that (1) holds in Definition 10 for each 1198751 119875
2isin prod(119883 120575)
satisfying condition (2) We take 119875 = ([119888119896 119889
119896] 119909
119896)119901
119894=1with
sum119896|119889
119896minus 119888
119896| lt 120578 and put 119875
1= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119862
119896 119889
119896ge
0) and 1198752= ([119888
119896 119889
119896] 119909
119896) 119865
119899(119888119896 119889
119896lt 0) So we have
|(cup1198751)Δ(cap119875
2)| = sum
119901
119896=1|119889
119896minus 119888
119896| lt 120578 Then by condition (1)
in Definition 10 we have
sup119899
119901
sum119896=1
119863(119865119899(119888119896) 119865
119899(119889
119896))
= sup119863( sum
(119888119896 119889119896)isin1198751
119865119899(119888119896 119889
119896)
sum(119888119896119889119896)isin1198752
119865119899(119888119896 119889
119896)) lt 120576
(16)
Hence we have that 119865 is uniformly 119860119862119866lowast
120575
We get the following theorem byTheorems 9 and 11
Theorem 12 Let the following conditions be satisfied
(i) 119891119899119883
rarr 119891119883ae in [119886 119887]where each119891
119899119883is strong fuzzy
Henstock integrable on [119886 119887](ii) the primitives 119865
119899119883of 119891
119899119883are119880119860119862nabla(119883)with closed set
119883 in [119886 119887]
Then 119891119883
is strong fuzzy Henstock integrable on [119886 119887] withprimitive 119865
119883
Next we give the controlled convergence theorem forthe strong fuzzy Henstock integrals by the definition of the119880119860119862119866
120575for a fuzzy-number-valued function
Definition 13 Let 119865 [119886 119887] rarr 119864119899 and let 119883 sub [119886 119887]A fuzzy-number-valued function 119865 is 119860119862
120575on 119883 if for each
120576 gt 0 there exist 120578 gt 0 and 120575(119909) gt 0 on 119883 such thatsum119873
119894=1119863(119865(119888
119894) 119865(119889
119894)) lt 120576 forsum119873
119894=1(119889
119894minus119888
119894) lt 120578 A fuzzy-number-
valued function 119865 is 119860119862119866120575on [119886 119887] if [119886 119887] is the union of a
sequence of set 119883119894 such that the function 119865 is 119860119862
120575(119883
119894) for
each 119894
Definition 14 (see [18]) A fuzzy-number-valued function 119865defined on 119883 sub [119886 119887] is said to be 119860119862lowast(119883) if for every 120576 gt 0
there exists 120578 gt 0 such that for every finite sequence of non-overlapping intervals [119886
119894 119887119894] satisfying sum119899
119894=1|119887119894minus 119886
119894| lt 120578
where 119886119894 119887119894isin 119883 for all 119894 one has sum120596(119865 [119886
119894 119887119894]) lt 120576 where 120596
denotes the oscillation of 119865 over [119886119894 119887119894] that is 120596(119865 [119886
119894 119887119894]) =
sup119863(119865(119910) 119865(119909)) 119909 119910 isin [119886119894 119887119894] A fuzzy-number-valued
function 119865 is said to be 119860119862119866lowast on 119883 if 119883 is the union ofa sequence of closed sets 119883
119894 such that on each 119883
119894 119865 is
119860119862lowast(119883119894)
Theorem 15 A fuzzy-number-valued function 119865 is 119860119862119866120575if
and only if it is 119860119862119866lowast on [119886 119887]
Theorem 16 (controlled convergence theorem) Let the fol-lowing conditions be satisfied
(1) 119891119899(119909) rarr 119891(119909) almost everywhere in [119886 119887] as 119899 rarr infin
where each 119891119899is strong fuzzy Henstock integrable on
[119886 119887]
Journal of Applied Mathematics 5
(2) the primitives 119865119899(119909) = (SFH) int119909
119886
119891119899(119904)d119909 of 119891
119899are
119880119860119862119866120575uniformly in 119899
(3) the sequence 119865119899(119909) converges uniformly to a con-
tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (17)
If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]
where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable
Proof By condition (2) there exists a sequence 119883119894 such that
119865119899isin 119880119860119862
120575in 119883
119894a bounded closed set with bounds 119886 and
119887 and put 119883 = 119883119894 We note that 119865
119899(119909) rarr 119865(119909) we have
119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866
120575on [119886 119887] ByTheorem 15
119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin
119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact
let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866
119899linearly
to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866
119899and119866 are119880119860119862 on [119886 119887] By condition
(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888
119896 119889
119896] be the intervals
contiguous to 119883 Then we have 119863(1198661015840
119899) le 119872
119896 We define a
fuzzy-number-valued function as follows
1198661015840
(119909) =119866119899(119889
119896) minus
119867119866119899(119888119896)
119889119896minus 119888
119896
119909 isin (119888119896 119889
119896) (18)
Consequently 1198661015840
119899(119909) converges on (119888
119896 119889
119896) Hence 1198661015840
119899con-
verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840
119899(119909) =
119891119899
rarr 119891 on 119883 Therefore we have 1198661015840
119899(119909) = 119892(119909) =
119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]
by Theorem 15 Therefore there exists an 119860119862119866120575function on
[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (19)
4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations
In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation
Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909
Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880
Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le
119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896
119899(119905) of step functions defined on |119905 minus 120591| le
119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)
almost everywhere as 119899 rarr infin Let 119865119899(119905) = int
119905
120591
119891(119904 119896119899(119904))d119904
Then 119865119899(119905) is 119880119860119862119866
120575uniformly in 119899 and equicontinuous
By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable
Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr
119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions
(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868
(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868
Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)
Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585
Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus
By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let
120601 (119905)=120595 (119905)+int119905
120591
119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
(2) the primitives 119865119899(119909) = (SFH) int119909
119886
119891119899(119904)d119909 of 119891
119899are
119880119860119862119866120575uniformly in 119899
(3) the sequence 119865119899(119909) converges uniformly to a con-
tinuous function on [119886 119887] Then 119891(119909) is strong fuzzyHenstock integrable on [119886 119887] and one has
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (17)
If conditions (1) and (2) are replaced by condition (4)(4) 119892(119909) le 119891(119909) le ℎ(119909) almost everywhere on [119886 119887]
where 119892(119909) and ℎ(119909) are strong fuzzy Henstock inte-grable
Proof By condition (2) there exists a sequence 119883119894 such that
119865119899isin 119880119860119862
120575in 119883
119894a bounded closed set with bounds 119886 and
119887 and put 119883 = 119883119894 We note that 119865
119899(119909) rarr 119865(119909) we have
119865 isin 119860119862120575on119883 and hence119865 isin 119860119862119866
120575on [119886 119887] ByTheorem 15
119865 isin 119860119862lowast on 119883 and hence 119865 isin 119860119862119866lowast on [119886 119887] and also 119865 isin
119860119862 on119883 and hence 119865 isin 119860119862119866 on [119886 119887]Now we prove that 1198651015840(119909) = 119891(119909) ae on [119886 119887] In fact
let 119866 [119886 119887] rarr 119864119899 equal 119865119899on 119883 and extend 119866
119899linearly
to the closed interval contiguous to119883 Likewise we define 119866from 119865 We see that119866
119899and119866 are119880119860119862 on [119886 119887] By condition
(3) we have 119866119899rarr 119866 on [119886 119887] Let [119888
119896 119889
119896] be the intervals
contiguous to 119883 Then we have 119863(1198661015840
119899) le 119872
119896 We define a
fuzzy-number-valued function as follows
1198661015840
(119909) =119866119899(119889
119896) minus
119867119866119899(119888119896)
119889119896minus 119888
119896
119909 isin (119888119896 119889
119896) (18)
Consequently 1198661015840
119899(119909) converges on (119888
119896 119889
119896) Hence 1198661015840
119899con-
verges on [119886 119887] ae Since 119866119899 isin 119860119862 on [119886 119887] then 1198661015840
119899(119909) =
119891119899
rarr 119891 on 119883 Therefore we have 1198661015840
119899(119909) = 119892(119909) =
119891(119909) = 1198651015840(119909) ae on 119883 Thus 1198651015840(119909) = 119891(119909) ae on [119886 119887]
by Theorem 15 Therefore there exists an 119860119862119866120575function on
[a 119887] such that 1198651015840(119909) = 119891(119909) ae on [119886 119887] Hence 119891 is strongfuzzy Henstock integrable on [119886 119887] and we have
lim119899rarrinfin
(SFH) int119887
119886
119891119899(119909) d119909 = (SFH) int
119887
119886
119891 (119909) d119909 (19)
4 The Generalized Solutions of DiscontinuousFuzzy Differential Equations
In this section a generalized fuzzy differential equation ofform (4) is defined by using strong fuzzy Henstock integralThemain results of this section are existence theorems for thegeneralized solution to the discontinuous fuzzy differentialequation
Definition 17 (see [11]) Let 120591 and 120585 be fixed and let a fuzzy-number-valued function 119891(119905 119909) be a Caratheodory functiondefined on a rectangle119880 |119905minus120591| le 119886 119863(119909 120585) le 119887 that is119891 iscontinuous in 119909 for almost all 119905 and measurable in 119905 for eachfixed 119909
Theorem 18 Let a fuzzy-number-valued function 119891 be afunction as given in Definition 17 then there exist two strongfuzzy Henstock integrable functions ℎ and 119892 defined on |119905minus120591| le119886 such that 119892(119905) le 119891(119905 119909) le ℎ(119905) for all (119905 119909) isin 119880
Proof Note that 119891 is a Caratheodory function Thus thereexist two measurable functions 119906(119905) and V(119905) defined on |119905 minus120591| le 119886 with values in 119863(119909 120585) le 119887 such that 119891(119905 119906(119905)) le
119891(119905 119909) le 119891(119905 V(119905)) for all (119905 119909) isin 119880 Next we will showthat 119891(119905 119906(119905)) and 119891(119905 V(119905)) are fuzzy Henstock integrable byusing controlled convergence Theorem 16 First there existsa sequence 119896
119899(119905) of step functions defined on |119905 minus 120591| le
119886 with values in 119863(119909 120585) le 119887 such that 119896119899(119905) rarr 119906(119905)
almost everywhere as 119899 rarr infin Let 119865119899(119905) = int
119905
120591
119891(119904 119896119899(119904))d119904
Then 119865119899(119905) is 119880119860119862119866
120575uniformly in 119899 and equicontinuous
By controlled convergence Theorem 16 119891(119905 119906(119905)) is strongfuzzy Henstock integrable Similarly 119891(119905 V(119905)) is strong fuzzyHenstock integrable
Definition 19 A fuzzy-number-valued function 119909(119905) 119868 rarr
119864119899 is said to be a solution of the discontinuous fuzzy differ-ential equation (4) if 119909(119905) satisfies the following conditions
(i) 119909(119905) is 119860119862119866120575on each compact subinterval of 119868
(ii) (119905 119909) isin 119880 for 119905 isin 119868(iii) 1199091015840(119905) for almost everywhere 119905 isin 119868
Now we will state the existence theorem for the general-ized solution of discontinuous fuzzy differential equation (4)
Theorem 20 Suppose that 119891 satisfies the condition ofTheorem 18 then there exists a generalized solution Φ of thediscontinuous fuzzy differential equation (4) on some interval|119905 minus 120591| le 119886 which satisfies Φ(120591) = 120585
Proof Given 119892(t) le 119891(119905 119909) le ℎ(119905) for all 119909 and almost all 119905with (119905 119909) isin 119880 we get 0 le 119891(119905 119909)minus
By Caratheodory existence theorem (see Theorem 7 in [11])there is a fuzzy-number-valued function 120595 on some interval|119905 minus 120591| le 119886 such that 1205951015840(119905) = 119865(119905 120595(119905)) almost everywhere inthis interval and 120595(120591) = 120585 Let
120601 (119905)=120595 (119905)+int119905
120591
119892 (119904) d119904 or 120601 (119905)=120595 (119905)+(minus1) sdot int119905
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
Example 21 Consider fuzzy differential equation 1199091015840 =
119891(119905 119909) = 119892(119905 119909) + ℎ(119905) where 119863(119892(119905 119909) 0) le 119863(1198921(119905) 0) for
all |119905| le 1 119863(119909 0) le 1 and 1198921(119905) is Kaleva integrable on |119905| le 1
and ℎ(119905) = 119860 sdot (119889119889119905)(1199052 sin 119905minus2) if 119905 = 0 and ℎ(0) = 0 Here 119860is defined in Example 1 Note that ℎ is strong fuzzy Henstockintegrable but not Kaleva integrable and
ℎ (119905) minus1198671198921(119905) le 119891 (119905 119909) le ℎ (119905) + 119892
1(119905)
for |119905| le 1 119863 (119909 0) le 1
(26)
Thus by Theorem 20 there exists a solution of 1199091015840 = 119891(119905 119909)
with 119909(0) = 0 For instance if 119892(119905 119909) = 1199052119909 then
120601 (119905) = 1198901199053
3
sdot int119905
0
119890minus1199043
3
ℎ (119904) d119904 (27)
is a solution by using integrating factor
We get the following existence theorem by Theorems 18and 20
Theorem 22 Let a fuzzy-number-valued function 119891 be aCaratheodory function defined on a rectangle 119880 |119905 minus
120591| le 119886119863(119909 120585) le 119887 Let 119891(119905 119906(119905)) be strong fuzzy Henstockintegrable on |119905 minus 120591| le 119886 for any step function 119906(119905) definedon |119905 minus 120591| le 119886 with values in 119863(119909 120585) le 119887 Denote that119865119906(119905) = int
119905
120591
119891(119904 119906(119904))d119904 If 119865119906 119906 is a step function is 119880119860119862119866
120575
uniformly in 119906 and equicontinuous on |119905 minus 120591| le 119886 then thereexists a solution 120601 of 1199091015840 = 119891(119905 119909) on some interval |119905 minus 120591| le 120573with 120601(120591) = 120585
Finally in this paper we will show the continuousdependence of a solution on parameters by using Theorems16 18 and 22
Let 119880119901be a connected set in 119880 Let 119888 gt 0 and let 120583
0be
fixed119868120583= 120583
1003816100381610038161003816120583 minus 12058301003816100381610038161003816 lt 119888
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
minus120583(119905) for 120583 isin (minus1 0] Let 119891(119905 119909 120583) = 1199052119909 + ℎ
120583(119905)
defined on (minus2 2) times (minus1 1) Then we have
119863(119865120583119906(1199051) 119865
120583119906(1199052)) = 119863(int
1199052
1199051
119891 (119904 119906 (119904) 120583) d119904 0)
le int1199052
1199051
1199042d119904 + 119863(int
1199052
1199051
ℎ120583d119904 0)
119863(int1199052
1199051
ℎ120583d119904 0) le int
1199052
1199051
119863(ℎ (119904) d119904 0)
(41)
where 1199051 1199052isin (0 1] or 119905
1 1199052isin [minus1 0) and
119863(int119905
0
ℎ120583(119904) d119904 0)
=
119863(119860 sdot 1199052 sin 119905minus2 119860 sdot 1205832 sin 120583minus2) 120583 = 0
119863 (119860 sdot 1199052 sin 119905minus2 0) 120583 = 0
(42)
Note that ℎ is Kaleva integrable on every subinterval of [minus1 0)and (0 1] Therefore we have that 119865
120583119906(119905) is equicontinuous
on [minus1 1] in 119906 and near 1205830= 0 Furthermore 119865
120583119906(119905) is
119860119862120575(119883
119899) uniformly in 120583 and 119906 where 119883
119899= [1119899 1] 119899 =
1 2 On the other hand by using integrating factor for119905 isin [minus1 1] we have
120601120583(119905) = 119890
1199052
3
int119905
0
119890minus1199043
3
119860 sdot ℎ120583(119904) d119904 (43)
with 120601120583(0) = 0 Obviously 120601
0is uniqueThus byTheorem 24
120601120583rarr 120601
0uniformly on [minus1 1]
5 Conclusion
In this paper we give the definition of the119880119860119862119866120575for a fuzzy-
number-valued function and the nonabsolute fuzzy integraland its controlled convergence theorem In addition we dealwith the Cauchy problem and the continuous dependence ofa solution on parameters of discontinuous fuzzy differentialequations involving the strong fuzzy Henstock integral infuzzy number space The function governing the equationsis supposed to be discontinuous with respect to some vari-ables and satisfy nonabsolute fuzzy integrability Our resultimproves the result given in [1 11 19 20] (where uniformcontinuity was required) as well as those referred therein
Acknowledgments
The authors are very grateful to the anonymous refereesand Professor Mehmet Sezer for many valuable commentsand suggestions which helped to improve the presentationof the paper The authors would like to thank the NationalNatural Science Foundation of China (no 11161041) and theFundamental Research Fund for the Central Universities (no31920130010)
8 Journal of Applied Mathematics
References
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008
[1] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987
[2] Z Gong and Y Shao ldquoGlobal existence and uniqueness ofsolutions for fuzzy differential equations under dissipative-typeconditionsrdquo Computers amp Mathematics with Applications vol56 no 10 pp 2716ndash2723 2008
[3] O Kaleva ldquoThe Cauchy problem for fuzzy differential equa-tionsrdquo Fuzzy Sets and Systems vol 35 no 3 pp 389ndash396 1990
[4] J J Nieto ldquoThe Cauchy problem for continuous fuzzy differen-tial equationsrdquo Fuzzy Sets and Systems vol 102 no 2 pp 259ndash262 1999
[5] M L Puri and D A Ralescu ldquoDifferentials of fuzzy functionsrdquoJournal of Mathematical Analysis and Applications vol 91 no 2pp 552ndash558 1983
[6] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987
[7] C Wu and S Song ldquoExistence theorem to the Cauchy problemof fuzzy differential equations under compactness-type condi-tionsrdquo Information Sciences vol 108 no 1ndash4 pp 123ndash134 1998
[8] C Wu S Song and Z Qi ldquoExistence and uniqueness for asolution on the closed subset to the Cauchy problem of fuzzydiffrential equationsrdquo Journal of Harbin Institute of Technologyvol 2 pp 1ndash7 1997
[9] CWu S Song and E S Lee ldquoApproximate solutions existenceand uniqueness of the Cauchy problem of fuzzy differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 202 no 2 pp 629ndash644 1996
[10] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions vol 6 Taylor amp FrancisLondon UK 2003
[11] X Xue and Y Fu ldquoCaratheodory solutions of fuzzy differentialequationsrdquo Fuzzy Sets and Systems vol 125 no 2 pp 239ndash2432002
[12] R HenstockTheory of Integration Butterworths London UK1963
[13] J Kurzweil ldquoGeneralized ordinary differential equations andcontinuous dependence on a parameterrdquo Czechoslovak Mathe-matical Journal vol 7 no 82 pp 418ndash449 1957
[14] P Y LeeLanzhou Lectures onHenstock Integration vol 2WorldScientific Publishing Teaneck NJ USA 1989
[15] C Wu and Z Gong ldquoOn Henstock integrals of interval-valuedfunctions and fuzzy-valued functionsrdquo Fuzzy Sets and Systemsvol 115 no 3 pp 377ndash391 2000
[16] C Wu and Z Gong ldquoOn Henstock integral of fuzzy-number-valued functions Irdquo Fuzzy Sets and Systems vol 120 no 3 pp523ndash532 2001
[17] Z Gong ldquoOn the problem of characterizing derivatives for thefuzzy-valued functions II Almost everywhere differentiabilityand strong Henstock integralrdquo Fuzzy Sets and Systems vol 145no 3 pp 381ndash393 2004
[18] Z Gong and Y Shao ldquoThe controlled convergence theoremsfor the strong Henstock integrals of fuzzy-number-valuedfunctionsrdquo Fuzzy Sets and Systems vol 160 no 11 pp 1528ndash1546 2009
[19] T S Chew and F Flordeliza ldquoOn 1199091015840 = 119891(119905 119909) and Henstock-Kurzweil integralsrdquo Differential and Integral Equations vol 4no 4 pp 861ndash868 1991
[20] B Bede and S G Gal ldquoGeneralizations of the differentiabilityof fuzzy-number-valued functions with applications to fuzzy
differential equationsrdquo Fuzzy Sets and Systems vol 151 no 3pp 581ndash599 2005
[21] D Dubois and H Prade ldquoTowards fuzzy differential calculus IIntegration of fuzzy mappingsrdquo Fuzzy Sets and Systems vol 8no 1 pp 1ndash17 1982
[22] P Diamond and P KloedenMetric Spaces of Fuzzy Sets WorldScientific Publishing River Edge NJ USA 1994
[23] R Goetschel Jr and W Voxman ldquoElementary fuzzy calculusrdquoFuzzy Sets and Systems vol 18 no 1 pp 31ndash43 1986
[24] Y Chalco-Cano and H Roman-Flores ldquoOn new solutions offuzzy differential equationsrdquo Chaos Solitons and Fractals vol38 no 1 pp 112ndash119 2008