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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 292643 9 pageshttpdxdoiorg1011552013292643
Research ArticleExtremal Solutions and Relaxation Problems forFractional Differential Inclusions
Juan J Nieto12 Abdelghani Ouahab3 and P Prakash14
1 Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela15782 Santiago de Compostela Spain
2Department of Mathematics King Abdulaziz University Jeddah 21589 Saudi Arabia3 Laboratory of Mathematics Sidi-Bel-Abbes University PO Box 89 22000 Sidi-Bel-Abbes Algeria4Department of Mathematics Periyar University Salem 636 011 India
Correspondence should be addressed to Juan J Nieto juanjosenietoroigusces
Received 10 May 2013 Accepted 31 July 2013
Academic Editor Daniel C Biles
Copyright copy 2013 Juan J Nieto et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present the existence of extremal solution and relaxation problem for fractional differential inclusion with initial conditions
1 Introduction
Differential equations with fractional order have recentlyproved to be valuable tools in the modeling of many physicalphenomena [1ndash9] There has also been a significant theoreti-cal development in fractional differential equations in recentyears see themonographs of Kilbas et al [10]Miller andRoss[11] Podlubny [12] and Samko et al [13] and the papers ofKilbas and Trujillo [14] Nahusev [15] Podlubny et al [16]and Yu and Gao [17]
Recently some basic theory for initial value problems forfractional differential equations and inclusions involving theRiemann-Liouville differential operator was discussed forexample by Lakshmikantham [18] and Chalco-Cano et al[19]
Applied problems requiring definitions of fractionalderivatives are those that are physically interpretable for ini-tial conditions containing 119910(0) 1199101015840
(0) and so forthThe samerequirements are true for boundary conditions Caputorsquosfractional derivative satisfies these demands Formore detailson the geometric and physical interpretation for fractionalderivatives of both Riemann-Liouville and Caputo types seePodlubny [12]
Fractional calculus has a long history We refer the readerto [20]
and inclusions with standard Riemann-Liouville and Caputoderivatives with differences conditions were studied byAbbaset al [21 22] Benchohra et al [23] Henderson and Ouahab[24 25] Jiao and Zhou [26] and Ouahab [27ndash29] and in thereferences therein
In this paper we will be concerned with the existence ofsolutions Filippovrsquos theorem and the relaxation theorem ofabstract fractional differential inclusions More precisely wewill consider the following problem
120572 is the Caputo fractional derivatives 120572 isin (1 2]119865 119869 times R119873
rarr P(R119873) is a multifunction and ext119865(119905 119910)
represents the set of extreme points of 119865(119905 119910) (P(R119873) is the
family of all nonempty subsets of R119873During the last couple of years the existence of extremal
solutions and relaxation problem for ordinary differentialinclusionswas studied bymany authors for example see [30ndash34] and the references therein
2 Abstract and Applied Analysis
The paper is organized as follows We first collect somebackground material and basic results from multivaluedanalysis and give some results on fractional calculus inSections 2 and 3 respectivelyThenwewill be concernedwiththe existence of solution for extremal problemThis is the aimof Section 4 In Section 5 we prove the relaxation problem
2 Preliminaries
The reader is assumed to be familiar with the theory of multi-valued analysis and differential inclusions in Banach spacesas presented in Aubin et al [35 36] Hu and Papageorgiou[37] Kisielewicz [38] and Tolstonogov [32]
Let (119883 sdot ) be a real Banach space [0 119887] an interval in 119877and 119862([0 119887] 119883) the Banach space of all continuous functionsfrom 119869 into119883 with the norm
1003817100381710038171003817 0 le 119905 le 119887 (3)
A measurable function 119910 [0 119887] rarr 119883 is Bochnerintegrable if 119910 is Lebesgue integrable In what follows1198711([0 119887] 119883) denotes the Banach space of functions 119910
[0 119887] rarr 119883 which are Bochner integrable with norm
100381710038171003817100381711991010038171003817100381710038171 = int
The norm sdot 119908is weaker than the usual norm sdot
1 and for a
broad class of subsets of 1198711([0 119887] 119883) the topology defined by
the weak norm coincides with the usual weak topology (see[37 Proposition 414 page 195]) Denote by
P (119883) = 119884 sub 119883 119884 = 0
Pcl (119883) = 119884 isin P (119883) 119884 closed
P119887(119883) = 119884 isin P (119883) 119884 bounded
Pcv (119883) = 119884 isin P (119883) 119884 convex
Pcp (119883) = 119884 isin P (119883) 119884 compact
(6)
A multivalued map 119866 119883 rarr P(119883) has convex (closed)values if 119866(119909) is convex (closed) for all 119909 isin 119883 We say that 119866is bounded on bounded sets if 119866(119861) is bounded in 119883 for eachbounded set 119861 of119883 (ie sup
Definition 1 A multifunction 119865 119883 rarr P(119884) is said to beupper semicontinuous at the point 119909
0isin 119883 if for every open
119882 sube 119884 such that 119865(1199090) sub 119882 there exists a neighborhood
119881(1199090) of 119909
0such that 119865(119881(119909
0)) sub 119882
A multifunction is called upper semicontinuous (usc forshort) on119883 if for each 119909 isin 119883 it is usc at 119909
Definition 2 A multifunction 119865 119883 rarr P(119884) is said to belower continuous at the point 119909
0isin 119883 if for every open119882 sube
119884 such that 119865(1199090)cap119882 = 0 there exists a neighborhood119881(119909
0)
of 1199090with property that 119865(119909) cap 119882 = 0 for all 119909 isin 119881(119909
0)
A multifunction is called lower semicontinuous (lsc forshort) provided that it is lower semicontinuous at every point119909 isin 119883
Lemma 3 (see [39 Lemma 32]) Let 119865 [0 119887] rarr P(119884)
be a measurable multivalued map and 119906 [119886 119887] rarr 119884 ameasurable function Then for any measurable V [119886 119887] rarr
(0 +infin) there exists a measurable selection 119891V of 119865 such thatfor ae 119905 isin [119886 119887]
1003817100381710038171003817119906 (119905) minus 119891V (119905)1003817100381710038171003817 le 119889 (119906 (119905) 119865 (119905)) + V (119905) (7)
First consider the Hausdorff pseudometric
119867119889 P (119864) timesP (119864) 997888rarr R
+cup infin (8)
defined by
119867119889(119860 119861) = maxsup
119886isin119860
119889 (119886 119861) sup119887isin119861
119889 (119860 119887) (9)
where 119889(119860 119887) = inf119886isin119860
119889(119886 119887) and 119889(119886 119861) = inf119887isin119861
119889(119886 119887)(P
119887cl(119864)119867119889) is a metric space and (Pcl(119883)119867119889
) is a gener-alized metric space
Definition 4 A multifunction 119865 119884 rarr P(119883) is calledHausdorff lower semicontinuous at the point 119910
0isin 119884 if for
any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910
0
such that
119865 (1199100) sub 119865 (119910) + 120598119861 (0 1) for every 119910 isin 119880 (119910
0) (10)
where 119861(0 1) is the unite ball in119883
Definition 5 A multifunction 119865 119884 rarr P(119883) is calledHausdorff upper semicontinuous at the point 119910
0isin 119884 if for
any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910
0
such that
119865 (119910) sub 119865 (1199100) + 120598119861 (0 1) for every 119910 isin 119880 (119910
0) (11)
119865 is called continuous if it is Hausdorff lower and uppersemicontinuous
Definition 6 Let 119883 be a Banach space a subset 119860 sub
1198711([0 119887] 119883) is decomposable if for all 119906 V isin 119860 and for every
Lebesgue measurable set 119868 sub 119869 one has
119906120594119868+ V120594
[0119887]119868isin 119860 (12)
where 120594119860stands for the characteristic function of the set 119860
We denote by Dco(1198711([0 119887] 119883)) the family of decomposable
sets
Abstract and Applied Analysis 3
Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871
1([0 119887] 119883)) defined by
F (119910) = V isin 1198711
([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))
ae 119905 isin [0 119887] (13)
The operator F is called the Nemytsrsquokiı operator associatedto 119865
Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values
Next we state a classical selection theorem due to Bressanand Colombo
Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873
119883 rarr P119888119897(119871
1([0 119887] 119864)) with closed decomposable values has
a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871
1([0 119887] 119864) such that 119891(119909) isin 119873(119909)
for every 119909 isin 119883
Let us introduce the following hypothesis
(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that
(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable
(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]
Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an
integrably bounded multivalued map satisfying (H1) Then 119865
Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly
compact in 1198711([0 119887] 119883)
Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)
possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869
120598sub [0 119887]whose Lebesgue
measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869
120598times 119884 rarr 119883 is
continuous with respect to the metric 119889119883(sdot sdot)
Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr
P119908cpcv(119883) is continuous with respect to 119910 for almost every
119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property
In what follows we present some definitions and proper-ties of extreme points
Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860
In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860
Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points
Let 1199091015840
119899119899isinN be a denumerable dense (in 120590(1198831015840
119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin
Pcpcv(119883) and 1199091015840
119899define the function
119889119899
(119860 119906) = max ⟨119910 minus 119911 1199091015840
119899⟩ 119910 119911 isin 119860 119906 =
119910 + 119911
2
(15)
Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0
for all 119899 ge 1
In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878
119865) is nonempty and co(ext(119878
119865)) = 119878
119865
Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a
measurable integrably bounded map Then
ext (119878119865) sube 119878
119865 (16)
where ext (119878119865) is the closure of set ext (119878
119865) in the topology of
the space 1198711([0 119887] 119883)
Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)
be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866
For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]
3 Fractional Calculus
According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads
119868119899119891 (119905) =
1
(119899 minus 1)int
119905
0
(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N
(19)
Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871
1([119886 119887]R) is defined by
119868120572
119886+119891 (119905) = int
119905
119886
(119905 minus 119904)120572minus1
Γ (120572)119891 (119904) 119889119904 (20)
where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601
120572(119905) where 120601
120572(119905) = 119905
(120572minus1)Γ(120572) for 119905 gt 0 and
we write 120601120572(119905) = 0 for 119905 le 0 and 120601
120572rarr 120575(119905) as 120572 rarr 0 where
120575 is the delta function and Γ is the Euler gamma functiondefined by
Γ (120572) = int
infin
0
119905120572minus1
119890minus119905119889119905 120572 gt 0 (21)
For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0
+ this limit may be infinite
After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that
119863119899119868119899= Id 119868
119899119863
119899= Id 119899 isin N (22)
that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that
119868119899119863
119899119891 (119905) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)(119886
+)(119905 minus 119886)
119896
119896 119905 gt 0 (23)
As a consequence we expect that 119863120572 is defined as the leftinverse to 119868
120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0
Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by
119863120572119891 (119905) =
1
Γ (119899 minus 120572)(119889
119889119905)
119899
int
119905
119886
(119905 minus 119904)minus120572+119899minus1
119891 (119904) 119889119904 (24)
where 119899 = [120572] + 1 and [120572] is the integer part of 120572
Defining for consistency 1198630= 119868
0= Id then we easily
recognize that
119863120572119868120572= Id 120572 ge 0 (25)
119863120572119905120574=
Γ (120574 + 1)
Γ (120574 + 1 minus 120572)119905120574minus120572
Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer
Note the remarkable fact that the fractional derivative119863
120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N
In fact (26) with 120574 = 0 illustrates that
1198631205721 =
(119905 minus 119886)minus120572
Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)
It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the
gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional
derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])
Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-
order derivative of 119891 is defined by
(119888
119863120572
119891) (119905) =1
Γ (119899 minus 120572)int
119905
119886
(119905 minus 119904)119899minus120572minus1
119891119899
(119904) 119889119904 (28)
This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572
lowastwe (tacitly) assume that this condition is
met We easily recognize that in general
119863120572119891 (119905) = 119863
119898119868119898minus120572
119891 (119905) = 119869119898minus120572
119863119898119891 (119905) = 119863
120572
lowast119891 (119905) (29)
unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0
119863120572119891 (119905) =
119888
119863120572
119891 (119905) +
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+) (30)
and therefore recalling the fractional derivative of the powerfunction (26) one has
119863120572(119891 (119905) minus
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+)) = 119863
120572
lowast119891 (119905) (31)
Abstract and Applied Analysis 5
The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886
+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero
119888119863
1205721 = 0 120572 gt 0 (32)
We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have
119863120572
(119905 minus 119886)120572minus119895
= 0 for 119895 = 1 2 [120572] + 1 (33)
From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
The paper is organized as follows We first collect somebackground material and basic results from multivaluedanalysis and give some results on fractional calculus inSections 2 and 3 respectivelyThenwewill be concernedwiththe existence of solution for extremal problemThis is the aimof Section 4 In Section 5 we prove the relaxation problem
2 Preliminaries
The reader is assumed to be familiar with the theory of multi-valued analysis and differential inclusions in Banach spacesas presented in Aubin et al [35 36] Hu and Papageorgiou[37] Kisielewicz [38] and Tolstonogov [32]
Let (119883 sdot ) be a real Banach space [0 119887] an interval in 119877and 119862([0 119887] 119883) the Banach space of all continuous functionsfrom 119869 into119883 with the norm
1003817100381710038171003817 0 le 119905 le 119887 (3)
A measurable function 119910 [0 119887] rarr 119883 is Bochnerintegrable if 119910 is Lebesgue integrable In what follows1198711([0 119887] 119883) denotes the Banach space of functions 119910
[0 119887] rarr 119883 which are Bochner integrable with norm
100381710038171003817100381711991010038171003817100381710038171 = int
The norm sdot 119908is weaker than the usual norm sdot
1 and for a
broad class of subsets of 1198711([0 119887] 119883) the topology defined by
the weak norm coincides with the usual weak topology (see[37 Proposition 414 page 195]) Denote by
P (119883) = 119884 sub 119883 119884 = 0
Pcl (119883) = 119884 isin P (119883) 119884 closed
P119887(119883) = 119884 isin P (119883) 119884 bounded
Pcv (119883) = 119884 isin P (119883) 119884 convex
Pcp (119883) = 119884 isin P (119883) 119884 compact
(6)
A multivalued map 119866 119883 rarr P(119883) has convex (closed)values if 119866(119909) is convex (closed) for all 119909 isin 119883 We say that 119866is bounded on bounded sets if 119866(119861) is bounded in 119883 for eachbounded set 119861 of119883 (ie sup
Definition 1 A multifunction 119865 119883 rarr P(119884) is said to beupper semicontinuous at the point 119909
0isin 119883 if for every open
119882 sube 119884 such that 119865(1199090) sub 119882 there exists a neighborhood
119881(1199090) of 119909
0such that 119865(119881(119909
0)) sub 119882
A multifunction is called upper semicontinuous (usc forshort) on119883 if for each 119909 isin 119883 it is usc at 119909
Definition 2 A multifunction 119865 119883 rarr P(119884) is said to belower continuous at the point 119909
0isin 119883 if for every open119882 sube
119884 such that 119865(1199090)cap119882 = 0 there exists a neighborhood119881(119909
0)
of 1199090with property that 119865(119909) cap 119882 = 0 for all 119909 isin 119881(119909
0)
A multifunction is called lower semicontinuous (lsc forshort) provided that it is lower semicontinuous at every point119909 isin 119883
Lemma 3 (see [39 Lemma 32]) Let 119865 [0 119887] rarr P(119884)
be a measurable multivalued map and 119906 [119886 119887] rarr 119884 ameasurable function Then for any measurable V [119886 119887] rarr
(0 +infin) there exists a measurable selection 119891V of 119865 such thatfor ae 119905 isin [119886 119887]
1003817100381710038171003817119906 (119905) minus 119891V (119905)1003817100381710038171003817 le 119889 (119906 (119905) 119865 (119905)) + V (119905) (7)
First consider the Hausdorff pseudometric
119867119889 P (119864) timesP (119864) 997888rarr R
+cup infin (8)
defined by
119867119889(119860 119861) = maxsup
119886isin119860
119889 (119886 119861) sup119887isin119861
119889 (119860 119887) (9)
where 119889(119860 119887) = inf119886isin119860
119889(119886 119887) and 119889(119886 119861) = inf119887isin119861
119889(119886 119887)(P
119887cl(119864)119867119889) is a metric space and (Pcl(119883)119867119889
) is a gener-alized metric space
Definition 4 A multifunction 119865 119884 rarr P(119883) is calledHausdorff lower semicontinuous at the point 119910
0isin 119884 if for
any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910
0
such that
119865 (1199100) sub 119865 (119910) + 120598119861 (0 1) for every 119910 isin 119880 (119910
0) (10)
where 119861(0 1) is the unite ball in119883
Definition 5 A multifunction 119865 119884 rarr P(119883) is calledHausdorff upper semicontinuous at the point 119910
0isin 119884 if for
any 120598 gt 0 there exists a neighbourhood 119880(1199100) of the point 119910
0
such that
119865 (119910) sub 119865 (1199100) + 120598119861 (0 1) for every 119910 isin 119880 (119910
0) (11)
119865 is called continuous if it is Hausdorff lower and uppersemicontinuous
Definition 6 Let 119883 be a Banach space a subset 119860 sub
1198711([0 119887] 119883) is decomposable if for all 119906 V isin 119860 and for every
Lebesgue measurable set 119868 sub 119869 one has
119906120594119868+ V120594
[0119887]119868isin 119860 (12)
where 120594119860stands for the characteristic function of the set 119860
We denote by Dco(1198711([0 119887] 119883)) the family of decomposable
sets
Abstract and Applied Analysis 3
Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871
1([0 119887] 119883)) defined by
F (119910) = V isin 1198711
([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))
ae 119905 isin [0 119887] (13)
The operator F is called the Nemytsrsquokiı operator associatedto 119865
Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values
Next we state a classical selection theorem due to Bressanand Colombo
Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873
119883 rarr P119888119897(119871
1([0 119887] 119864)) with closed decomposable values has
a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871
1([0 119887] 119864) such that 119891(119909) isin 119873(119909)
for every 119909 isin 119883
Let us introduce the following hypothesis
(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that
(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable
(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]
Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an
integrably bounded multivalued map satisfying (H1) Then 119865
Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly
compact in 1198711([0 119887] 119883)
Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)
possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869
120598sub [0 119887]whose Lebesgue
measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869
120598times 119884 rarr 119883 is
continuous with respect to the metric 119889119883(sdot sdot)
Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr
P119908cpcv(119883) is continuous with respect to 119910 for almost every
119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property
In what follows we present some definitions and proper-ties of extreme points
Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860
In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860
Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points
Let 1199091015840
119899119899isinN be a denumerable dense (in 120590(1198831015840
119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin
Pcpcv(119883) and 1199091015840
119899define the function
119889119899
(119860 119906) = max ⟨119910 minus 119911 1199091015840
119899⟩ 119910 119911 isin 119860 119906 =
119910 + 119911
2
(15)
Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0
for all 119899 ge 1
In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878
119865) is nonempty and co(ext(119878
119865)) = 119878
119865
Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a
measurable integrably bounded map Then
ext (119878119865) sube 119878
119865 (16)
where ext (119878119865) is the closure of set ext (119878
119865) in the topology of
the space 1198711([0 119887] 119883)
Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)
be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866
For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]
3 Fractional Calculus
According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads
119868119899119891 (119905) =
1
(119899 minus 1)int
119905
0
(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N
(19)
Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871
1([119886 119887]R) is defined by
119868120572
119886+119891 (119905) = int
119905
119886
(119905 minus 119904)120572minus1
Γ (120572)119891 (119904) 119889119904 (20)
where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601
120572(119905) where 120601
120572(119905) = 119905
(120572minus1)Γ(120572) for 119905 gt 0 and
we write 120601120572(119905) = 0 for 119905 le 0 and 120601
120572rarr 120575(119905) as 120572 rarr 0 where
120575 is the delta function and Γ is the Euler gamma functiondefined by
Γ (120572) = int
infin
0
119905120572minus1
119890minus119905119889119905 120572 gt 0 (21)
For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0
+ this limit may be infinite
After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that
119863119899119868119899= Id 119868
119899119863
119899= Id 119899 isin N (22)
that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that
119868119899119863
119899119891 (119905) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)(119886
+)(119905 minus 119886)
119896
119896 119905 gt 0 (23)
As a consequence we expect that 119863120572 is defined as the leftinverse to 119868
120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0
Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by
119863120572119891 (119905) =
1
Γ (119899 minus 120572)(119889
119889119905)
119899
int
119905
119886
(119905 minus 119904)minus120572+119899minus1
119891 (119904) 119889119904 (24)
where 119899 = [120572] + 1 and [120572] is the integer part of 120572
Defining for consistency 1198630= 119868
0= Id then we easily
recognize that
119863120572119868120572= Id 120572 ge 0 (25)
119863120572119905120574=
Γ (120574 + 1)
Γ (120574 + 1 minus 120572)119905120574minus120572
Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer
Note the remarkable fact that the fractional derivative119863
120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N
In fact (26) with 120574 = 0 illustrates that
1198631205721 =
(119905 minus 119886)minus120572
Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)
It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the
gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional
derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])
Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-
order derivative of 119891 is defined by
(119888
119863120572
119891) (119905) =1
Γ (119899 minus 120572)int
119905
119886
(119905 minus 119904)119899minus120572minus1
119891119899
(119904) 119889119904 (28)
This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572
lowastwe (tacitly) assume that this condition is
met We easily recognize that in general
119863120572119891 (119905) = 119863
119898119868119898minus120572
119891 (119905) = 119869119898minus120572
119863119898119891 (119905) = 119863
120572
lowast119891 (119905) (29)
unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0
119863120572119891 (119905) =
119888
119863120572
119891 (119905) +
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+) (30)
and therefore recalling the fractional derivative of the powerfunction (26) one has
119863120572(119891 (119905) minus
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+)) = 119863
120572
lowast119891 (119905) (31)
Abstract and Applied Analysis 5
The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886
+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero
119888119863
1205721 = 0 120572 gt 0 (32)
We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have
119863120572
(119905 minus 119886)120572minus119895
= 0 for 119895 = 1 2 [120572] + 1 (33)
From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
Let 119865 [0 119887] times 119883 rarr P(119883) be a multivalued map withnonempty closed values Assign to119865 themultivalued operatorF 119862([0 119887] 119883) rarr P(119871
1([0 119887] 119883)) defined by
F (119910) = V isin 1198711
([0 119887] 119883) V (119905) isin 119865 (119905 119910 (119905))
ae 119905 isin [0 119887] (13)
The operator F is called the Nemytsrsquokiı operator associatedto 119865
Definition 7 Let 119865 [0 119887] times 119883 rarr P(119883) be a multivaluedmapwith nonempty compact valuesWe say that119865 is of lowersemicontinuous type (lsc type) if its associated Nemytsrsquokiıoperator F is lower semicontinuous and has nonemptyclosed and decomposable values
Next we state a classical selection theorem due to Bressanand Colombo
Lemma 8 (see [40]) Let119883 be a separable metric space and let119864 be a Banach spaceThen every lsc multivalued operator119873
119883 rarr P119888119897(119871
1([0 119887] 119864)) with closed decomposable values has
a continuous selection that is there exists a continuous single-valued function 119891 119883 rarr 119871
1([0 119887] 119864) such that 119891(119909) isin 119873(119909)
for every 119909 isin 119883
Let us introduce the following hypothesis
(H1) 119865 [0 119887]times119883 rarr P(119883) is a nonempty compact valuedmultivalued map such that
(a) the mapping (119905 119910) 997891rarr 119865(119905 119910) is L otimes Bmeasurable
(b) the mapping 119910 997891rarr 119865(119905 119910) is lower semicontinu-ous for ae 119905 isin [0 119887]
Lemma 9 (see eg [41]) Let 119865 119869 times 119883 rarr P119888119901(119864) be an
integrably bounded multivalued map satisfying (H1) Then 119865
Lemma 10 (see [37]) Let 119870 sub 119883 be a weakly compactsubset of 119883 Then 119865(119870) is relatively weakly compact subset of1198711([0 119887] 119883) Moreover if 119870 is convex then 119865(119870) is weakly
compact in 1198711([0 119887] 119883)
Definition 11 A multifunction 119865 [0 119887] times 119884 rarr P119908cpcv(119883)
possesses the Scorza-Dragoni property (S-D property) if foreach 120598 gt 0 there exists a closed set 119869
120598sub [0 119887]whose Lebesgue
measure 120583(119869120598) le 120598 and such that 119865 [0 119887] 119869
120598times 119884 rarr 119883 is
continuous with respect to the metric 119889119883(sdot sdot)
Remark 12 It is well known that if the map 119865 [0 119887] times 119884 rarr
P119908cpcv(119883) is continuous with respect to 119910 for almost every
119905 isin [0 119887] and is measurable with respect to 119905 for every 119910 isin 119884then it possesses the S-D property
In what follows we present some definitions and proper-ties of extreme points
Definition 13 Let119860 be a nonempty subset of a real or complexlinear vector space An extreme point of a convex set 119860 is apoint 119909 isin 119860 with the property that if 119909 = 120582119910 + (1 minus 120582)119911 with119910 119911 isin 119860 and 120582 isin [0 1] then 119910 = 119909 andor 119911 = 119909 ext(119860) willdenote the set of extreme points of 119860
In other words an extreme point is a point that is not aninterior point of any line segment lying entirely in 119860
Lemma 14 (see [42]) A nonempty compact set in a locallyconvex linear topological space has extremal points
Let 1199091015840
119899119899isinN be a denumerable dense (in 120590(1198831015840
119883) topol-ogy) subset of the set 119909 isin 119883 119909 le 1 For any 119860 isin
Pcpcv(119883) and 1199091015840
119899define the function
119889119899
(119860 119906) = max ⟨119910 minus 119911 1199091015840
119899⟩ 119910 119911 isin 119860 119906 =
119910 + 119911
2
(15)
Lemma 15 (see [33]) 119906 isin ext(119860) if and only if 119889119899(119860 119906) = 0
for all 119899 ge 1
In accordance with Krein-Milman and Trojansky theo-rem [43] the set ext(119878
119865) is nonempty and co(ext(119878
119865)) = 119878
119865
Lemma 16 (see [33]) Let 119865 [0 119887] rarr P119908119888119901119888V(119883) be a
measurable integrably bounded map Then
ext (119878119865) sube 119878
119865 (16)
where ext (119878119865) is the closure of set ext (119878
119865) in the topology of
the space 1198711([0 119887] 119883)
Theorem 17 (see [33]) Let 119865 [0 119887] times 119884 rarr P119908119888119901119888V(119883)
be a multivalued map that has the 119878-119863 property and let it beintegrable bounded on compacts from 119884 Consider a compactsubset 119870 sub 119862([0 119887] 119883) and define the multivalued map 119866
For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]
3 Fractional Calculus
According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads
119868119899119891 (119905) =
1
(119899 minus 1)int
119905
0
(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N
(19)
Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871
1([119886 119887]R) is defined by
119868120572
119886+119891 (119905) = int
119905
119886
(119905 minus 119904)120572minus1
Γ (120572)119891 (119904) 119889119904 (20)
where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601
120572(119905) where 120601
120572(119905) = 119905
(120572minus1)Γ(120572) for 119905 gt 0 and
we write 120601120572(119905) = 0 for 119905 le 0 and 120601
120572rarr 120575(119905) as 120572 rarr 0 where
120575 is the delta function and Γ is the Euler gamma functiondefined by
Γ (120572) = int
infin
0
119905120572minus1
119890minus119905119889119905 120572 gt 0 (21)
For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0
+ this limit may be infinite
After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that
119863119899119868119899= Id 119868
119899119863
119899= Id 119899 isin N (22)
that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that
119868119899119863
119899119891 (119905) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)(119886
+)(119905 minus 119886)
119896
119896 119905 gt 0 (23)
As a consequence we expect that 119863120572 is defined as the leftinverse to 119868
120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0
Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by
119863120572119891 (119905) =
1
Γ (119899 minus 120572)(119889
119889119905)
119899
int
119905
119886
(119905 minus 119904)minus120572+119899minus1
119891 (119904) 119889119904 (24)
where 119899 = [120572] + 1 and [120572] is the integer part of 120572
Defining for consistency 1198630= 119868
0= Id then we easily
recognize that
119863120572119868120572= Id 120572 ge 0 (25)
119863120572119905120574=
Γ (120574 + 1)
Γ (120574 + 1 minus 120572)119905120574minus120572
Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer
Note the remarkable fact that the fractional derivative119863
120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N
In fact (26) with 120574 = 0 illustrates that
1198631205721 =
(119905 minus 119886)minus120572
Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)
It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the
gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional
derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])
Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-
order derivative of 119891 is defined by
(119888
119863120572
119891) (119905) =1
Γ (119899 minus 120572)int
119905
119886
(119905 minus 119904)119899minus120572minus1
119891119899
(119904) 119889119904 (28)
This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572
lowastwe (tacitly) assume that this condition is
met We easily recognize that in general
119863120572119891 (119905) = 119863
119898119868119898minus120572
119891 (119905) = 119869119898minus120572
119863119898119891 (119905) = 119863
120572
lowast119891 (119905) (29)
unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0
119863120572119891 (119905) =
119888
119863120572
119891 (119905) +
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+) (30)
and therefore recalling the fractional derivative of the powerfunction (26) one has
119863120572(119891 (119905) minus
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+)) = 119863
120572
lowast119891 (119905) (31)
Abstract and Applied Analysis 5
The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886
+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero
119888119863
1205721 = 0 120572 gt 0 (32)
We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have
119863120572
(119905 minus 119886)120572minus119895
= 0 for 119895 = 1 2 [120572] + 1 (33)
From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
For a background of extreme point of 119865(119905 119910(119905)) seeDunford-Schwartz [42 Chapter 5 Section 8] and Florenzanoand Le Van [44 Chapter 3]
3 Fractional Calculus
According to the Riemann-Liouville approach to fractionalcalculus the notation of fractional integral of order 120572 (120572 gt 0)is a natural consequence of the well known formula (usuallyattributed to Cauchy) that reduces the calculation of the119899-fold primitive of a function 119891(119905) to a single integral ofconvolution type In our notation the Cauchy formula reads
119868119899119891 (119905) =
1
(119899 minus 1)int
119905
0
(119905 minus 119904)119899minus1119891 (119904) 119889119904 119905 gt 0 119899 isin N
(19)
Definition 18 (see [13 45]) The fractional integral of order120572 gt 0 of a function 119891 isin 119871
1([119886 119887]R) is defined by
119868120572
119886+119891 (119905) = int
119905
119886
(119905 minus 119904)120572minus1
Γ (120572)119891 (119904) 119889119904 (20)
where Γ is the gamma function When 119886 = 0 we write119868120572119891(119905) = 119891(119905)lowast120601
120572(119905) where 120601
120572(119905) = 119905
(120572minus1)Γ(120572) for 119905 gt 0 and
we write 120601120572(119905) = 0 for 119905 le 0 and 120601
120572rarr 120575(119905) as 120572 rarr 0 where
120575 is the delta function and Γ is the Euler gamma functiondefined by
Γ (120572) = int
infin
0
119905120572minus1
119890minus119905119889119905 120572 gt 0 (21)
For consistency 1198680 = Id (identity operator) that is 1198680119891(119905) =119891(119905) Furthermore by 119868120572119891(0+) we mean the limit (if it exists)of 119868120572119891(119905) for 119905 rarr 0
+ this limit may be infinite
After the notion of fractional integral that of fractionalderivative of order 120572 (120572 gt 0) becomes a natural requirementand one is attempted to substitute 120572 with minus120572 in the aboveformulas However this generalization needs some care inorder to guarantee the convergence of the integral andpreserve the well known properties of the ordinary derivativeof integer order Denoting by119863119899 with 119899 isin N the operator ofthe derivative of order 119899 we first note that
119863119899119868119899= Id 119868
119899119863
119899= Id 119899 isin N (22)
that is119863119899 is the left inverse (and not the right inverse) to thecorresponding integral operator 119869119899 We can easily prove that
119868119899119863
119899119891 (119905) = 119891 (119905) minus
119899minus1
sum
119896=0
119891(119896)(119886
+)(119905 minus 119886)
119896
119896 119905 gt 0 (23)
As a consequence we expect that 119863120572 is defined as the leftinverse to 119868
120572 For this purpose introducing the positiveinteger 119899 such that 119899 minus 1 lt 120572 le 119899 one defines the fractionalderivative of order 120572 gt 0
Definition 19 For a function 119891 given on interval [119886 119887] the120572th Riemann-Liouville fractional-order derivative of 119891 isdefined by
119863120572119891 (119905) =
1
Γ (119899 minus 120572)(119889
119889119905)
119899
int
119905
119886
(119905 minus 119904)minus120572+119899minus1
119891 (119904) 119889119904 (24)
where 119899 = [120572] + 1 and [120572] is the integer part of 120572
Defining for consistency 1198630= 119868
0= Id then we easily
recognize that
119863120572119868120572= Id 120572 ge 0 (25)
119863120572119905120574=
Γ (120574 + 1)
Γ (120574 + 1 minus 120572)119905120574minus120572
Of course properties (25) and (26) are a natural generaliza-tion of those known when the order is a positive integer
Note the remarkable fact that the fractional derivative119863
120572119891 is not zero for the constant function 119891(119905) = 1 if 120572 notin N
In fact (26) with 120574 = 0 illustrates that
1198631205721 =
(119905 minus 119886)minus120572
Γ (1 minus 120572) 120572 gt 0 119905 gt 0 (27)
It is clear that 1198631205721 = 0 for 120572 isin N due to the poles of the
gamma function at the points 0 minus1 minus2 We now observe an alternative definition of fractional
derivative originally introduced by Caputo [46 47] in thelate sixties and adopted by Caputo and Mainardi [48] inthe framework of the theory of Linear Viscoelasticity (see areview in [4])
Definition 20 Let 119891 isin 119860119862119899([119886 119887]) The Caputo fractional-
order derivative of 119891 is defined by
(119888
119863120572
119891) (119905) =1
Γ (119899 minus 120572)int
119905
119886
(119905 minus 119904)119899minus120572minus1
119891119899
(119904) 119889119904 (28)
This definition is of course more restrictive thanRiemann-Liouville definition in that it requires the absoluteintegrability of the derivative of order 119898 Whenever we usethe operator 119863120572
lowastwe (tacitly) assume that this condition is
met We easily recognize that in general
119863120572119891 (119905) = 119863
119898119868119898minus120572
119891 (119905) = 119869119898minus120572
119863119898119891 (119905) = 119863
120572
lowast119891 (119905) (29)
unless the function 119891(119905) along with its first 119899 minus 1 derivativesvanishes at 119905 = 119886+ In fact assuming that the passage of the119898-derivative under the integral is legitimate we recognize thatfor119898 minus 1 lt 120572 lt 119898 and 119905 gt 0
119863120572119891 (119905) =
119888
119863120572
119891 (119905) +
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+) (30)
and therefore recalling the fractional derivative of the powerfunction (26) one has
119863120572(119891 (119905) minus
119898minus1
sum
119896=0
(119905 minus 119886)119896minus120572
Γ (119896 minus 120572 + 1)119891
(119896)(119886
+)) = 119863
120572
lowast119891 (119905) (31)
Abstract and Applied Analysis 5
The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886
+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero
119888119863
1205721 = 0 120572 gt 0 (32)
We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have
119863120572
(119905 minus 119886)120572minus119895
= 0 for 119895 = 1 2 [120572] + 1 (33)
From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
The alternative definition that is Definition 20 for thefractional derivative thus incorporates the initial values of thefunction and of lower order The subtraction of the Taylorpolynomial of degree 119899 minus 1 at 119905 = 119886
+ from 119891(119905) means a sortof regularization of the fractional derivative In particularaccording to this definition the relevant property for whichthe fractional derivative of a constant is still zero
119888119863
1205721 = 0 120572 gt 0 (32)
We now explore the most relevant differences between thetwo fractional derivatives given in Definitions 19 and 20From Riemann-Liouville fractional derivatives we have
119863120572
(119905 minus 119886)120572minus119895
= 0 for 119895 = 1 2 [120572] + 1 (33)
From (32) and (33) we thus recognize the following state-ments about functions which for 119905 gt 0 admit the samefractional derivative of order 120572 with 119899 minus 1 lt 120572 le 119899 119899 isin N
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
This proves that 119873 is continuous Hence by Schauderrsquos fixedpoint there exists 119910 isin 119870 such that 119910 = 119873(119910)
5 The Relaxed Problem
In this section we examine whether the solutions of theextremal problem are dense in those of the convexified oneSuch a result is important in optimal control theory whetherthe relaxed optimal state can be approximated by originalstates the relaxed problems are generally much simpler tobuild For the problem for first-order differential inclusionswe refer for example to [35 Theorem 2 page 124] or [36Theorem 1044 page 402] For the relaxation of extremalproblems we see the following recent references [30 50]
Now we present our main result of this section
Theorem 25 Let 119865 [0 119887] times R119873rarr P(R119873
) be amultifunction satisfying the following hypotheses
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
The multivalued map 119905 rarr 119865(119905 sdot) is measurable and 119909 rarr
119865(sdot 119909) is 119867119889continuous In addition if 119865(sdot sdot) has compact
values then 119865(sdot sdot) is graph measurable and the mapping119905 rarr 119865(119905 119910(119905)) is a measurable multivalued map for fixed 119910 isin119862([0 119887]R119873
) Then by Lemma 3 there exists a measurableselection V
1(119905) isin 119865(119905 119910(119905)) ae 119905 isin [0 119887] such that
Since the measurable multifunction 119865 is integrable boundedLemma 9 implies that the Nemytsrsquokiı operatorF has decom-posable values Hence 119910 rarr 119866
120598(119910) is lsc with decomposable
values By Lemma 8 there exists a continuous selection 119891120598
119862([0 119887]R119873) rarr 119871
1(119869R119873
) such that
119891120598(119910) isin 119866
120598(119910) forall119910 isin 119862 ([0 119887] R
119873) (61)
FromTheorem 17 there exists function 119892120598 119870 rarr 119871
rarr R119873 are Caratheodory functionsand bounded
Then (2) is solvable
Example 27 If in addition to the conditions on 119865 ofExample 26 119891
1and 119891
2are Lipschitz functions then 119878
119890= 119878
119888
Acknowledgments
This work is partially supported by the Ministerio de Econo-mia y Competitividad Spain project MTM2010-15314 andcofinanced by the European Community Fund FEDER
8 Abstract and Applied Analysis
References
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
[1] K Diethelm and A D Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelingof viscoplasticityrdquo in Scientifice Computing in Chemical Engi-neering II-Computational Fluid Dynamics Reaction Engineeringand Molecular Properties F Keil W Mackens H Voss andJ Werther Eds pp 217ndash224 Springer Heidelberg Germany1999
[2] L Gaul P Klein and S Kemple ldquoDamping descriptioninvolving fractional operatorsrdquo Mechanical Systems and SignalProcessing vol 5 no 2 pp 81ndash88 1991
[3] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995
[4] F Mainardi ldquoFractional calculus some basic problems incontinuum and statistical mechanicsrdquo in Fractals and Frac-tional Calculus in Continuum Mechanics A Carpinteri and FMainardi Eds pp 291ndash348 Springer Wien Germany 1997
[5] A B Malinowska and D F M Torres ldquoTowards a combinedfractionalmechanics and quantizationrdquo Fractional Calculus andApplied Analysis vol 15 no 3 pp 407ndash417 2012
[6] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995
[7] S P Nasholm and S Holm ldquoOn a fractional Zener elastic waveequationrdquo Fractional Calculus and Applied Analysis vol 16 no1 pp 26ndash50 2013
[8] L Vazquez J J Trujillo and M P Velasco ldquoFractional heatequation and the second law of thermodynamicsrdquo FractionalCalculus and Applied Analysis vol 14 no 3 pp 334ndash342 2011
[9] B J West and D West ldquoFractional dynamics of allometryrdquoFractional Calculus and Applied Analysis vol 15 no 1 pp 70ndash96 2012
[10] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherland 2006
[11] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New York NYUSA 1993
[12] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[13] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and DerivativesTheory and Applications Gordon andBreach Science Yverdon Switzerland 1993
[14] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002
[15] A M Nahusev ldquoThe Sturm-Liouville problem for a secondorder ordinary differential equation with fractional derivativesin the lower termsrdquoDoklady Akademii Nauk SSSR vol 234 no2 pp 308ndash311 1977
[16] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002
[17] C Yu and G Gao ldquoExistence of fractional differential equa-tionsrdquo Journal of Mathematical Analysis and Applications vol310 no 1 pp 26ndash29 2005
[18] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquo Nonlinear Analysis Theory Methods amp Applica-tions A vol 69 no 10 pp 3337ndash3343 2008
[19] Y Chalco-Cano J J Nieto A Ouahab and H Roman-FloresldquoSolution set for fractional differential equationswithRiemann-Liouville derivativerdquo Fractional Calculus and Applied Analysisvol 16 no 3 pp 682ndash694 2013
[20] J T Machado V Kiryakova and FMainardi ldquoRecent history offractional calculusrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 3 pp 1140ndash1153 2011
[21] S Abbas and M Benchohra ldquoFractional order Riemann-Liouville integral inclusions with two independent variablesand multiple delayrdquo Opuscula Mathematica vol 33 no 2 pp209ndash222 2013
[22] S Abbas M Benchohra and J J Nieto ldquoGlobal uniquenessresults for fractional order partial hyperbolic functional differ-ential equationsrdquo Advances in Difference Equations vol 2011Article ID 379876 25 pages 2011
[23] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional functional differential inclu-sions with infinite delay and applications to control theoryrdquoFractional Calculus amp Applied Analysis vol 11 no 1 pp 35ndash562008
[24] J Henderson and A Ouahab ldquoFractional functional differentialinclusions with finite delayrdquo Nonlinear Analysis Theory Meth-ods amp Applications A vol 70 no 5 pp 2091ndash2105 2009
[25] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010
[26] F Jiao and Y Zhou ldquoExistence of solutions for a class offractional boundary value problems via critical point theoryrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1181ndash1199 2011
[27] A Ouahab ldquoSome results for fractional boundary value prob-lemof differential inclusionsrdquoNonlinear AnalysisTheoryMeth-ods amp Applications A vol 69 no 11 pp 3877ndash3896 2008
[28] A Ouahab ldquoFilippovrsquos theorem for impulsive differential inclu-sions with fractional orderrdquo Electronic Journal of QualitativeTheory of Differential Equations no 23 pp 1ndash23 2009
[29] AOuahab ldquoFractional semilinear differential inclusionsrdquoCom-puters amp Mathematics with Applications vol 64 no 10 pp3235ndash3252 2012
[30] F S de Blasi and G Pianigiani ldquoNon-convex valued differentialinclusions in Banach spacesrdquo Journal of Mathematical Analysisand Applications vol 128 pp 541ndash555 1996
[31] N S Papageorgiou ldquoOn the ldquobang-bangrdquo principle for nonlin-ear evolution inclusionsrdquo Aequationes Mathematicae vol 45no 2-3 pp 267ndash280 1993
[32] A Tolstonogov Differential Inclusions in a Banach SpaceKluwer Academic Dordrecht The Netherlands 2000
[33] A A Tolstonogov ldquoExtreme continuous selectors of multi-valued maps and their applicationsrdquo Journal of DifferentialEquations vol 122 no 2 pp 161ndash180 1995
[34] A A Tolstonogov ldquoExtremal selectors ofmultivaluedmappingsand the ldquobangbangrdquo principle for evolution inclusionsrdquo Dok-lady Akademii Nauk SSSR vol 317 no 3 pp 589ndash593 1991(Russian) translation in Soviet Mathematics Doklady vol 43no 2 481ndash485 1991
[35] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
Abstract and Applied Analysis 9
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010
[36] J-P Aubin andH Frankowska Set-Valued Analysis BirkhauserBoston Mass USA 1990
[37] S Hu and N S Papageorgiou Handbook of Multivalued Anal-ysis Volume I Theory Kluwer Academic Publishers LondonUK 1997
[38] M Kisielewicz Differential Inclusions and Optimal ControlKluwer Academic Dordrecht The Netherlands 1991
[39] Q J Zhu ldquoOn the solution set of differential inclusions inBanach spacerdquo Journal of Differential Equations vol 93 no 2pp 213ndash237 1991
[40] A Bressan andG Colombo ldquoExtensions and selections ofmapswith decomposable valuesrdquo Studia Mathematica vol 90 no 1pp 70ndash85 1988
[41] M Frigon and A Granas ldquoTheoremes drsquoexistence pour desinclusions differentielles sans convexiterdquo Comptes Rendus delrsquoAcademie des Sciences Serie I vol 310 no 12 pp 819ndash822 1990
[42] N Dunford and J T Schwartz Linear Operators I GeneralTheoryWith theAssistance ofWG Bade andR G Bartle Pureand Applied Mathematics Interscience Publishers New YorkNY USA 1958
[43] R D BourginGeometric Aspects of Convex Sets with the Radon-Nikodyrsquom Property vol 993 of Lecture Notes in MathematicsSpringer Berlin Germany 1983
[44] M Florenzano andC LeVan Finite Dimensional Convexity andOptimization vol 13 of Studies in Economic Theory SpringerBerlin Germany 2001 In Cooperation with Pascal Gourdel
[45] S Abbas M Benchohra and G M NrsquoGuerekata Topics inFractional Differential Equations vol 27 of Developments inMathematics Springer New York NY USA 2012
[46] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969
[47] M Caputo ldquoLinear models of dissipation whose Q is almostfrequency independent part IIrdquoGeophysical Journal of the RoyalAstronomical Society vol 13 pp 529ndash529 1967
[48] M Caputo and F Mainardi ldquoLinear models of dissipation inanelastic solidsrdquo La Rivista del Nuovo Cimento vol 1 no 2 pp161ndash198 1971
[49] M Benamara Point extremaux multi-applications et fonction-nelles integrales [These de 3eme Cycle] Universite de GrenobleGrenoble France 1975
[50] F S de Blasi and G Pianigiani ldquoBairersquos category and the bang-bang property for evolution differential inclusions of contrac-tive typerdquo Journal of Mathematical Analysis and Applicationsvol 367 no 2 pp 550ndash567 2010
[51] B Ahmad and J J Nieto ldquoA study of impulsive fractionaldifferential inclusions with anti-periodic boundary conditionsrdquoFractional Differential Calculus vol 2 no 1 pp 1ndash15 2012
[52] B Ahmad and J J Nieto ldquoExistence results for higher orderfractional differential inclusions with nonlocal boundary con-ditionsrdquo Nonlinear Studies vol 17 no 2 pp 131ndash138 2010