Extremal solutions and relaxation problems for fractional differential inclusions

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 292643, 9pages http://dx.doi.org/10.1155/2013/292643 Research Article Extremal Solutions and Relaxation Problems for Fractional Differential Inclusions Juan J. Nieto,1,2 Abdelghani Ouahab,3and P. Prakash1,4 1Departamento de An´ alisis Matematico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain 2Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3Laboratory of Mathematics, Sidi-Bel-Abb` es University, P.O. Box 89, 22000 Sidi-Bel-Abb` es, Algeria 4Department of Mathematics, Periyar University, Salem 636 011, India Correspondence should be addressed to Juan J. Nieto; juanjose.nieto[email protected] Received 10 May 2013; Accepted 31 July 2013 Academic Editor: Daniel C. Biles Copyright © 2013 Juan J. Nieto et al. This is an open access article distributed under the Creative Commons Attribution License, which 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 recently proved to be valuable tools in the modeling of many physical phenomena [1–9]. There has also been a significant theoretical development in fractional differential equations in recent years; see the monographs of Kilbas et al. [10], Miller and Ross [11], Podlubny [12], and Samko et al. [13]andthepapersof Kilbas and Trujillo [14], Nahuˇ sev [15], Podlubny et al. [16], and Yu and Gao [17]. Recently, some basic theory for initial value problems for fractional differential equations and inclusions involving the Riemann-Liouville differential operator was discussed, for example, by Lakshmikantham [18]andChalco-Canoetal. [19]. Applied problems requiring definitions of fractional derivatives are those that are physically interpretable for initial conditions containing 𝑦(0),𝑦󸀠(0), and so forth. The same requirements are true for boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both Riemann-Liouville and Caputo types, see Podlubny [12]. Fractional calculus has a long history. We refer the reader to [20]. Recently fractional functional differential equations and inclusions and impulsive fractional differential equations and inclusions with standard Riemann-Liouville and Caputo derivatives with differences conditions were studied by Abbas et al. [21,22], Benchohra et al. [23], Henderson and Ouahab [24,25], Jiao and Zhou [26], and Ouahab [27–29]andinthe references therein. In this paper, we will be concerned with the existence of solutions, Filippov’s theorem, and the relaxation theorem of abstract fractional differential inclusions. More precisely, we will consider the following problem: 𝑐𝐷𝛼𝑦(𝑡)∈𝐹(𝑡,𝑦(𝑡)), a.e.𝑡∈𝐽:=[0,𝑏], 𝑦(0)=𝑦0,𝑦 󸀠(0)=𝑦1,(1) 𝑐𝐷𝛼𝑦(𝑡)∈ext 𝐹(𝑡,𝑦(𝑡)), a.e.𝑡∈𝐽:=[0,𝑏], 𝑦(0)=𝑦0,𝑦 󸀠(0)=𝑦1,(2) where 𝑐𝐷𝛼is the Caputo fractional derivatives, 𝛼 ∈ (1,2], 𝐹:𝐽×R𝑁→P(R𝑁)is a multifunction, and ext 𝐹(𝑡,𝑦) represents the set of extreme points of 𝐹(𝑡,𝑦).(P(R𝑁)is the family of all nonempty subsets of R𝑁. During the last couple of years, the existence of extremal solutions and relaxation problem for ordinary differential inclusions was studied by many authors, for example, see [30– 34] and the references therein. 2Abstract and Applied Analysis The paper is organized as follows. We first collect some background material and basic results from multivalued analysis and give some results on fractional calculus in Sections2and 3, respectively. Then, we will be concerned with the existence of solution for extremal problem. This is the aim of Section 4.InSection 5, we prove the relaxation problem. 2. Preliminaries The reader is assumed to be familiar with the theory of multivalued analysis and differential inclusions in Banach spaces, as presented in Aubin et al. [35,36], Hu and Papageorgiou [37], Kisielewicz [38], and Tolstonogov [32]. Let (𝑋,‖⋅‖)be a real Banach space, [0,𝑏]an interval in 𝑅, and 𝐶([0,𝑏],𝑋)theBanachspaceofallcontinuousfunctions from 𝐽into 𝑋with the norm 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩∞=sup {󵄩󵄩󵄩󵄩𝑦(𝑡)󵄩󵄩󵄩󵄩:0≤𝑡≤𝑏}.(3) A measurable function 𝑦:[0,𝑏]→𝑋is Bochner integrable if ‖𝑦‖ is Lebesgue integrable. In what follows, 𝐿1([0,𝑏],𝑋) denotes the Banach space of functions 𝑦: [0,𝑏] → 𝑋, which are Bochner integrable with norm 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩1=∫𝑏 0󵄩󵄩󵄩󵄩𝑦(𝑡)󵄩󵄩󵄩󵄩𝑑𝑡. (4) Denote by 𝐿1 𝑤([0,𝑏],𝑋)the space of equivalence classes of Bochner integrable function 𝑦:[0,𝑏]→𝑋with the norm 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩𝑤=sup 𝑡∈[0,𝑡] 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0𝑦(𝑠)𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩.(5) The norm ‖⋅‖𝑤is weaker than the usual norm ‖⋅‖1,andfora broad class of subsets of 𝐿1([0,𝑏],𝑋),thetopologydefinedby the weak norm coincides with the usual weak topology (see [37,Proposition4.14,page195]).Denoteby P(𝑋)={𝑌⊂𝑋:𝑌 =0}, Pcl (𝑋)={𝑌∈P(𝑋):𝑌closed}, P𝑏(𝑋)={𝑌∈P(𝑋):𝑌bounded}, Pcv (𝑋)={𝑌∈P(𝑋):𝑌convex}, Pcp (𝑋)={𝑌∈P(𝑋):𝑌compact}. (6) Amultivaluedmap𝐺:𝑋→P(𝑋)has convex (closed) values if 𝐺(𝑥)is convex (closed) for all 𝑥∈𝑋.Wesaythat𝐺 is bounded on bounded sets if 𝐺(𝐵)is bounded in 𝑋for each bounded set 𝐵of 𝑋(i.e., sup𝑥∈𝐵{sup{‖𝑦‖:𝑦∈𝐺(𝑥)}}<∞). Definition 1. A multifunction 𝐹:𝑋→P(𝑌)is said to be uppersemicontinuousatthepoint𝑥0∈𝑋,if,foreveryopen 𝑊⊆𝑌such that 𝐹(𝑥0)⊂𝑊, there exists a neighborhood 𝑉(𝑥0)of 𝑥0such that 𝐹(𝑉(𝑥0))⊂𝑊. A multifunction is called upper semicontinuous (u.s.c. for short) on 𝑋if for each 𝑥∈𝑋it is u.s.c. at 𝑥. Definition 2. A multifunction 𝐹:𝑋→P(𝑌)is said to be lower continuous at the point 𝑥0∈𝑋,if,foreveryopen𝑊⊆ 𝑌such that 𝐹(𝑥0)∩𝑊 =0, there exists a neighborhood 𝑉(𝑥0) of 𝑥0with property that 𝐹(𝑥)∩𝑊 =0for all 𝑥∈𝑉(𝑥0). A multifunction is called lower semicontinuous (l.s.c. for short) provided that it is lower semicontinuous at every point 𝑥∈𝑋. Lemma 3 (see [39, Lemma 3.2]). Let 𝐹:[0,𝑏]→P(𝑌) be a measurable multivalued map and 𝑢:[𝑎,𝑏]→𝑌a measurablefunction.ThenforanymeasurableV: [𝑎,𝑏] → (0,+∞), there exists a measurable selection 𝑓Vof 𝐹such that for a.e. 𝑡∈[𝑎,𝑏], 󵄩󵄩󵄩󵄩𝑢(𝑡)−𝑓V(𝑡)󵄩󵄩󵄩󵄩≤𝑑(𝑢(𝑡),𝐹(𝑡))+V(𝑡).(7) First, consider the Hausdorff pseudometric 𝐻𝑑:P(𝐸)×P(𝐸)󳨀→ R+∪{∞},(8) defined by 𝐻𝑑(𝐴,𝐵)=max {sup 𝑎∈𝐴 𝑑(𝑎,𝐵),sup 𝑏∈𝐵 𝑑(𝐴,𝑏)}, (9) where 𝑑(𝐴,𝑏) = inf𝑎∈𝐴𝑑(𝑎,𝑏)and 𝑑(𝑎,𝐵) = inf𝑏∈𝐵𝑑(𝑎,𝑏). (P𝑏,cl(𝐸),𝐻𝑑)is a metric space and (Pcl(𝑋),𝐻𝑑)is a generalized metric space. Definition 4. A multifunction 𝐹:𝑌→P(𝑋)is called Hausdorff lower semicontinuous at the point 𝑦0∈𝑌,iffor any 𝜖>0there exists a neighbourhood 𝑈(𝑦0)of the point 𝑦0 such that 𝐹(𝑦0)⊂𝐹(𝑦)+𝜖𝐵(0,1),for every 𝑦∈𝑈(𝑦0),(10) where 𝐵(0,1)is the unite ball in 𝑋. Definition 5. A multifunction 𝐹:𝑌→P(𝑋)is called Hausdorff upper semicontinuous at the point 𝑦0∈𝑌,iffor any 𝜖>0there exists a neighbourhood 𝑈(𝑦0)of the point 𝑦0 such that 𝐹(𝑦)⊂𝐹(𝑦0)+𝜖𝐵(0,1),for every 𝑦∈𝑈(𝑦0). (11) 𝐹is called continuous, if it is Hausdorff lower and upper semicontinuous. Definition 6. Let 𝑋be a Banach space; a subset 𝐴⊂ 𝐿1([0,𝑏],𝑋)is decomposable if, for all 𝑢,V∈𝐴and for every Lebesgue measurable set 𝐼⊂𝐽,onehas 𝑢𝜒𝐼+V𝜒[0,𝑏]\𝐼 ∈𝐴, (12) where 𝜒𝐴stands for the characteristic function of the set 𝐴. We denote by Dco(𝐿1([0,𝑏],𝑋))the family of decomposable sets. Abstract and Applied Analysis 3 Let 𝐹:[0,𝑏]×𝑋→P(𝑋)be a multivalued map with nonempty closed values. Assign to 𝐹the multivalued operator F:𝐶([0,𝑏],𝑋)→P(𝐿1([0,𝑏],𝑋))defined by F(𝑦)={V∈𝐿1([0,𝑏],𝑋):V(𝑡)∈𝐹(𝑡,𝑦(𝑡)), a.e.𝑡∈[0,𝑏]}. (13) The operator Fis called the Nemyts’ki˘ ı operator associated to 𝐹. Definition 7. Let 𝐹:[0,𝑏]×𝑋→P(𝑋)be a multivalued mapwithnonemptycompactvalues.Wesaythat𝐹is of lower semicontinuous type (l.s.c. type) if its associated Nemyts’ki˘ ı operator Fis lower semicontinuous and has nonempty closed and decomposable values. Next, we state a classical selection theorem due to Bressan and Colombo. Lemma 8 (see [40]). Let 𝑋be a separable metric space and let 𝐸be a Banach space. Then every l.s.c. multivalued operator 𝑁: 𝑋→P𝑐𝑙(𝐿1([0,𝑏],𝐸))with closed decomposable values has a continuous selection; that is, there exists a continuous singlevalued function 𝑓:𝑋→𝐿1([0,𝑏],𝐸)such that 𝑓(𝑥)∈𝑁(𝑥) for every 𝑥∈𝑋. Let us introduce the following hypothesis. (H1)𝐹:[0,𝑏]×𝑋→P(𝑋)is a nonempty compact valued multivalued map such that (a) the mapping (𝑡,𝑦) 󳨃→ 𝐹(𝑡,𝑦) is L⊗B measurable; (b) the mapping 𝑦󳨃→𝐹(𝑡,𝑦)is lower semicontinuous for a.e. 𝑡∈[0,𝑏]. Lemma 9 (see, e.g., [41]). Let 𝐹:𝐽×𝑋→P𝑐𝑝(𝐸)be an integrably bounded multivalued map satisfying (H1).Then𝐹 is of lower semicontinuous type. Define 𝐹(𝐾)={𝑓∈𝐿1([0,𝑏],𝑋):𝑓(𝑡)∈𝐾a.e.𝑡∈[0,𝑏]}, 𝐾⊂𝑋,(14) where 𝑋is a Banach space. Lemma 10 (see [37]). Let 𝐾⊂𝑋be a weakly compact subset of 𝑋.Then𝐹(𝐾)is relatively weakly compact subset of 𝐿1([0,𝑏],𝑋).Moreoverif𝐾is convex, then 𝐹(𝐾)is weakly compact in 𝐿1([0,𝑏],𝑋). Definition 11. A multifunction 𝐹:[0,𝑏]×𝑌→P𝑤cpcv(𝑋) possesses the Scorza-Dragoni property (S-D property) if for each 𝜖>0, there exists a closed set 𝐽𝜖⊂[0,𝑏]whose Lebesgue measure 𝜇(𝐽𝜖)≤𝜖and such that 𝐹:[0,𝑏]\𝐽𝜖×𝑌→𝑋is continuous with respect to the metric 𝑑𝑋(⋅,⋅). Remark 12. Itiswellknownthatifthemap𝐹:[0,𝑏]×𝑌→ P𝑤cpcv(𝑋)is continuous with respect to 𝑦for almost every 𝑡∈[0,𝑏]andismeasurablewithrespectto𝑡for every 𝑦∈𝑌, then it possesses the S-D property. In what follows, we present some definitions and properties of extreme points. Definition 13. Let 𝐴beanonemptysubsetofarealorcomplex linear vector space. An extreme point of a convex set 𝐴is a point 𝑥∈𝐴with the property that if 𝑥=𝜆𝑦+(1−𝜆)𝑧with 𝑦,𝑧∈𝐴and 𝜆∈[0,1],then𝑦=𝑥and/or 𝑧=𝑥. ext(𝐴)will denote the set of extreme points of 𝐴. In other words, an extreme point is a point that is not an interior point of any line segment lying entirely in 𝐴. Lemma 14 (see [42]). A nonempty compact set in a locally convex linear topological space has extremal points. Let {𝑥󸀠 𝑛}𝑛∈Nbe a denumerable, dense (in 𝜎(𝑋󸀠,𝑋)topology) subset of the set {𝑥∈𝑋:‖𝑥‖≤1}.Forany𝐴∈ Pcpcv(𝑋)and 𝑥󸀠 𝑛define the function 𝑑𝑛(𝐴,𝑢)=max {⟨𝑦−𝑧,𝑥󸀠 𝑛⟩:𝑦,𝑧∈𝐴,𝑢=𝑦+𝑧 2}.(15) Lemma 15 (see [33]). 𝑢∈ext(𝐴)if and only if 𝑑𝑛(𝐴,𝑢)=0 for all 𝑛≥1. In accordance with Krein-Milman and Trojansky theorem [43], the set ext(𝑆𝐹)is nonempty and co(ext(𝑆𝐹))=𝑆𝐹. Lemma 16 (see [33]). Let 𝐹:[0,𝑏]→P𝑤𝑐𝑝𝑐V(𝑋)be a measurable, integrably bounded map. Then ext (𝑆𝐹)⊆𝑆𝐹,(16) where ext (𝑆𝐹)is the closure of set ext (𝑆𝐹)in the topology of the space 𝐿1([0,𝑏],𝑋). Theorem 17 (see [33]). Let 𝐹:[0,𝑏]×𝑌→P𝑤𝑐𝑝𝑐V(𝑋) be a multivalued map that has the 𝑆-𝐷property and let it be integrable bounded on compacts from 𝑌.Consideracompact subset 𝐾⊂𝐶([0,𝑏],𝑋)and define the multivalued map 𝐺: 𝐾→𝐿 1([0,𝑏],𝑋),by 𝐺(𝑦(⋅)) ={𝑓∈𝐿1([0,𝑏],𝑋):𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡)) 𝑎.𝑒.𝑜𝑛 [0,𝑏]}, 𝑦∈𝐾. (17) Then for every 𝐾compact in 𝐶([0,𝑏],𝑋),𝜖>0and any continuous selection 𝑓:𝐾→𝐿 1([0,𝑏],𝑋), there exists a continuous selector 𝑔:𝐾→𝐿1([0,𝑏],𝑋)of the map ext (𝐺): 𝐾→𝐿 1([0,𝑏],𝑋)such that for all 𝑦∈𝐶([0,𝑏],𝑋)one has sup 𝑡∈[0,𝑏] 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0((𝑓𝑦)(𝑠)−(𝑔𝑦)(𝑠))𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩≤𝜖. (18) 4Abstract and Applied Analysis For a background of extreme point of 𝐹(𝑡,𝑦(𝑡)) see Dunford-Schwartz [42, Chapter 5, Section 8] and Florenzano andLeVan[44,Chapter3]. 3. Fractional Calculus According to the Riemann-Liouville approach to fractional calculus, the notation of fractional integral of order 𝛼(𝛼>0) is a natural consequence of the well known formula (usually attributed to Cauchy) that reduces the calculation of the 𝑛-fold primitive of a function 𝑓(𝑡) to a single integral of convolution type. In our notation the Cauchy formula reads 𝐼𝑛𝑓(𝑡):= 1 (𝑛−1)!∫𝑡 0(𝑡−𝑠)𝑛−1𝑓(𝑠)𝑑𝑠, 𝑡>0, 𝑛∈N. (19) Definition 18 (see [13,45]). The fractional integral of order 𝛼>0of a function 𝑓∈𝐿1([𝑎,𝑏],R)is defined by 𝐼𝛼 𝑎+𝑓(𝑡)=∫𝑡 𝑎(𝑡−𝑠)𝛼−1 Γ(𝛼)𝑓(𝑠)𝑑𝑠, (20) where Γis the gamma function. When 𝑎=0,wewrite 𝐼𝛼𝑓(𝑡)=𝑓(𝑡)∗𝜙𝛼(𝑡),where𝜙𝛼(𝑡)=𝑡(𝛼−1)/Γ(𝛼)for 𝑡>0,and we write 𝜙𝛼(𝑡)=0for 𝑡≤0and 𝜙𝛼→ 𝛿(𝑡)as 𝛼→0,where 𝛿is the delta function and Γis the Euler gamma function defined by Γ(𝛼)=∫∞ 0𝑡𝛼−1𝑒−𝑡𝑑𝑡, 𝛼>0. (21) For consistency, 𝐼0=Id (identity operator), that is, 𝐼0𝑓(𝑡)= 𝑓(𝑡).Furthermore,by𝐼𝛼𝑓(0+)we mean the limit (if it exists) of 𝐼𝛼𝑓(𝑡)for 𝑡→0 +; this limit may be infinite. After the notion of fractional integral, that of fractional derivative of order 𝛼(𝛼>0)becomes a natural requirement and one is attempted to substitute 𝛼with −𝛼in the above formulas. However, this generalization needs some care in order to guarantee the convergence of the integral and preserve the well known properties of the ordinary derivative of integer order. Denoting by 𝐷𝑛,with𝑛∈N, the operator of the derivative of order 𝑛,wefirstnotethat 𝐷𝑛𝐼𝑛=Id,𝐼 𝑛𝐷𝑛=Id,𝑛∈N,(22) that is, 𝐷𝑛istheleftinverse(andnottherightinverse)tothe corresponding integral operator 𝐽𝑛.Wecaneasilyprovethat 𝐼𝑛𝐷𝑛𝑓(𝑡)=𝑓(𝑡)−𝑛−1 ∑ 𝑘=0𝑓(𝑘) (𝑎+)(𝑡−𝑎)𝑘 𝑘! ,𝑡>0.(23) As a consequence, we expect that 𝐷𝛼is defined as the left inverse to 𝐼𝛼. For this purpose, introducing the positive integer 𝑛such that 𝑛−1<𝛼≤𝑛, one defines the fractional derivative of order 𝛼>0. Definition 19. For a function 𝑓givenoninterval[𝑎,𝑏],the 𝛼th Riemann-Liouville fractional-order derivative of 𝑓is defined by 𝐷𝛼𝑓(𝑡)=1 Γ(𝑛−𝛼)(𝑑 𝑑𝑡)𝑛∫𝑡 𝑎(𝑡−𝑠)−𝛼+𝑛−1𝑓(𝑠)𝑑𝑠, (24) where 𝑛=[𝛼]+1and [𝛼]is the integer part of 𝛼. Defining for consistency, 𝐷0=𝐼 0=Id, then we easily recognize that 𝐷𝛼𝐼𝛼=Id,𝛼≥0, (25) 𝐷𝛼𝑡𝛾=Γ(𝛾+1) Γ(𝛾+1−𝛼)𝑡𝛾−𝛼, 𝛼>0, 𝛾∈(−1,0)∪(0,+∞),𝑡>0. (26) Of course, properties (25)and(26)areanaturalgeneralization of those known when the order is a positive integer. Note the remarkable fact that the fractional derivative 𝐷𝛼𝑓is not zero for the constant function 𝑓(𝑡)=1,if𝛼∉N. In fact, (26)with𝛾=0illustrates that 𝐷𝛼1=(𝑡−𝑎)−𝛼 Γ(1−𝛼), 𝛼>0,𝑡>0. (27) It is clear that 𝐷𝛼1=0,for𝛼∈N, due to the poles of the gamma function at the points 0,−1,−2,.... We now observe an alternative definition of fractional derivative, originally introduced by Caputo [46,47]inthe late sixties and adopted by Caputo and Mainardi [48]in the framework of the theory of Linear Viscoelasticity (see a review in [4]). Definition 20. Let 𝑓∈𝐴𝐶 𝑛([𝑎,𝑏]). The Caputo fractionalorder derivative of 𝑓is defined by (𝑐𝐷𝛼𝑓)(𝑡):= 1 Γ(𝑛−𝛼)∫𝑡 𝑎(𝑡−𝑠)𝑛−𝛼−1𝑓𝑛(𝑠)𝑑𝑠. (28) This definition is of course more restrictive than Riemann-Liouville definition, in that it requires the absolute integrability of the derivative of order 𝑚.Wheneverweuse the operator 𝐷𝛼 ∗we (tacitly) assume that this condition is met. We easily recognize that in general 𝐷𝛼𝑓(𝑡):=𝐷𝑚𝐼𝑚−𝛼𝑓(𝑡)=𝐽𝑚−𝛼𝐷𝑚𝑓(𝑡):=𝐷𝛼 ∗𝑓(𝑡),(29) unless the function 𝑓(𝑡), along with its first 𝑛−1derivatives, vanishes at 𝑡=𝑎+. In fact, assuming that the passage of the 𝑚derivative under the integral is legitimate, we recognize that, for 𝑚−1<𝛼<𝑚and 𝑡>0, 𝐷𝛼𝑓(𝑡)=𝑐𝐷𝛼𝑓(𝑡)+𝑚−1 ∑ 𝑘=0 (𝑡−𝑎)𝑘−𝛼 Γ(𝑘−𝛼+1)𝑓(𝑘) (𝑎+), (30) and therefore, recalling the fractional derivative of the power function (26), one has 𝐷𝛼(𝑓(𝑡)−𝑚−1 ∑ 𝑘=0 (𝑡−𝑎)𝑘−𝛼 Γ(𝑘−𝛼+1)𝑓(𝑘) (𝑎+))=𝐷𝛼 ∗𝑓(𝑡).(31) Abstract and Applied Analysis 5 The alternative definition, that is, Definition 20,forthe fractional derivative thus incorporates the initial values of the function and of lower order. The subtraction of the Taylor polynomial of degree 𝑛−1at 𝑡=𝑎+from 𝑓(𝑡)means a sort of regularization of the fractional derivative. In particular, according to this definition, the relevant property for which the fractional derivative of a constant is still zero: 𝑐𝐷𝛼1=0, 𝛼>0. (32) We now explore the most relevant differences between the two fractional derivatives given in Definitions 19 and 20. From Riemann-Liouville fractional derivatives, we have 𝐷𝛼(𝑡−𝑎)𝛼−𝑗 =0, for 𝑗=1,2,...,[𝛼]+1. (33) From (32)and(33)wethusrecognizethefollowingstatements about functions which, for 𝑡>0,admitthesame fractional derivative of order 𝛼,with𝑛−1<𝛼≤𝑛,𝑛∈N: 𝐷𝛼𝑓(𝑡)=𝐷𝛼𝑔(𝑡)⇐⇒ 𝑓(𝑡)=𝑔(𝑡)+𝑚 ∑ 𝑗=1𝑐𝑗(𝑡−𝑎)𝛼−𝑗, 𝑐𝐷𝛼𝑓(𝑡)=𝑐𝐷𝛼𝑔(𝑡)⇐⇒ 𝑓(𝑡)=𝑔(𝑡)+𝑚 ∑ 𝑗=1𝑐𝑗(𝑡−𝑎)𝑛−𝑗. (34) In these formulas, the coefficients 𝑐𝑗are arbitrary constants. For proving all main results we present the following auxiliary lemmas. Lemma 21 (see [10]). Let 𝛼>0and let 𝑦∈𝐿 ∞(𝑎,𝑏)or 𝐶([𝑎,𝑏]).Then (𝑐𝐷𝛼𝐼𝛼𝑦)(𝑡)=𝑦(𝑡).(35) Lemma 22 (see [10]). Let 𝛼>0and 𝑛=[𝛼]+1.If𝑦∈ 𝐴𝐶𝑛[𝑎,𝑏]or 𝑦∈𝐶𝑛[𝑎,𝑏],then (𝐼𝛼𝑐𝐷𝛼𝑦)(𝑡)=𝑦(𝑡)−𝑛−1 ∑ 𝑘=0𝑦(𝑘) (𝑎) 𝑘! (𝑡−𝑎)𝑘.(36) For further readings and details on fractional calculus, we refer to the books and papers by Kilbas [10], Podlubny [12], Samko [13],andCaputo[46–48]. 4. Existence Result Definition 23. Afunction𝑦 ∈ 𝐶([0,𝑏],R𝑁)is called mild solution of problem (1) if there exist 𝑓∈𝐿1(𝐽,R𝑁)such that 𝑦(𝑡)=𝑦0+𝑡𝑦1+1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)1−𝛼𝑓(𝑠)𝑑𝑠, 𝑡∈[0,𝑏], (37) where 𝑓∈𝑆𝐹,𝑦 ={V∈𝐿1([0,𝑏],R𝑁):𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e. on [0,𝑏]}. We will impose the following conditions on 𝐹. (H1)The function 𝐹:𝐽×R𝑁→Pcpcv(R𝑁)such that (a) for all 𝑥∈R𝑁,themap𝑡 󳨃→ 𝐹(𝑡,𝑥) is measurable, (b) for every 𝑡∈[0,𝑏],themultivaluedmap𝑥→ 𝐹(𝑡,𝑥)is 𝐻𝑑continuous (H2)There exist 𝑝∈𝐿 1(𝐽,R+)and a continuous nondecreasing function 𝜓:[0,∞) → (0,∞)such that ‖𝐹(𝑡,𝑥)‖P=sup {‖V‖:V∈𝐹(𝑡,𝑥)}≤𝑝(𝑡)𝜓(‖𝑥‖), for a.e.𝑡∈[0,𝑏]and each 𝑥∈R𝑁,(38) with ∫𝑏 0𝑝(𝑠)𝑑𝑠<∫∞ ‖𝑦0‖+𝑏‖𝑦1‖𝑑𝑢 𝜓(𝑢).(39) Theorem 24. Assume that the conditions (H1)-(H2)and then the problem (2)have at least one solution. Proof. From (H2)there exists 𝑀>0such that ‖𝑦‖∞≤𝑀 for each 𝑦∈𝑆𝑐. Let 𝐹1(𝑡,𝑦)={ { { { { 𝐹(𝑡,𝑦) if 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩≤𝑀, 𝐹(𝑡,𝑀𝑦 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩)if 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩≥𝑀. (40) We consider 𝑐𝐷𝛼𝑦(𝑡)∈𝐹1(𝑡,𝑦(𝑡)), a.e.𝑡∈[0,𝑏], 𝑦(0)=𝑦0,𝑦 󸀠(0)=𝑦1.(41) It is clear that all the solutions of (41)aresolutionsof(2). Set 𝑉={𝑓∈𝐿1([0,𝑏],R𝑁):󵄩󵄩󵄩󵄩𝑓(𝑡)󵄩󵄩󵄩󵄩≤𝜓∗(𝑡)}, 𝜓∗(𝑡)=𝑝(𝑡)𝜓(𝑀).(42) It is clear that 𝑉is weakly compact in 𝐿1([0,𝑏],R𝑁).Remark that for every 𝑓∈𝑉, there exists a unique solution 𝐿(𝑓)of the following problem: 𝑐𝐷𝛼𝑦(𝑡)=𝑓(𝑡),a.e.𝑡∈[0,𝑏], 𝑦(𝑡)=𝑦0,𝑦 󸀠(0)=𝑦1;(43) this solution is defined by 𝐿(𝑓)(𝑡)=𝑦0+𝑡𝑦1+1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑓(𝑠)𝑑𝑠, a.e.𝑡∈[0,𝑏].(44) We claim that 𝐿is continuous. Indeed, let 𝑓𝑛→𝑓converge in 𝐿1([0,𝑏],R𝑁),as𝑛→∞,set𝑦𝑛=𝐿(𝑓𝑛),𝑛∈N.Itisclear 6Abstract and Applied Analysis that {𝑦𝑛:𝑛∈N}is relatively compact in 𝐶([0,𝑏],R𝑁)and 𝑦𝑛 converge to 𝑦∈𝐶([0,𝑏],R𝑁).Let 𝑧(𝑡)=𝑦0+𝑦1𝑡+ 1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑓(𝑠)𝑑𝑠, 𝑡∈[0,𝑏]. (45) Then 󵄩󵄩󵄩󵄩𝑦𝑛−𝑧󵄩󵄩󵄩󵄩∞≤𝑏𝛼 Γ(𝛼)∫𝑏 0󵄩󵄩󵄩󵄩𝑓𝑛(𝑠)−𝑓(𝑠)󵄩󵄩󵄩󵄩𝑑𝑠󳨀→0, as 𝑛󳨀→∞. (46) Hence 𝐾=𝐿(𝑉)is compact and convex subset of 𝐶([0,𝑏],R𝑁).Let𝑆𝐹:𝐾→Pclcv(𝐿1([0,𝑏],R𝑁))be the multivalued Nemitsky operator defined by 𝑆𝐹1(𝑦)={𝑓∈𝐿1([0,𝑏],R𝑁):𝑓(𝑡)∈𝐹1(𝑡,𝑦(𝑡)), a.e.𝑡∈[0,𝑏]}:=𝑆𝐹1,𝑦.(47) It is clear that 𝐹1(⋅,⋅) is 𝐻𝑑continuous and 𝐹1(⋅,⋅) ∈ P𝑤𝑘cpcv(R𝑁)and is integrably bounded, then by Theorem 17 (see also Theorem 6.5 in [32]orTheorem1.1in[34]), we can find a continuous function 𝑔:𝐾→𝐿1 𝑤([0,𝑏],R𝑁)such that 𝑔(𝑥)∈ext 𝑆𝐹1(𝑦) ∀𝑦∈𝐾. (48) From Benamara [49]weknowthat ext 𝑆𝐹1(𝑦)=𝑆ext 𝐹1(⋅,𝑦(⋅)) ∀𝑦∈𝐾. (49) Setting 𝑁=𝐿∘𝑔and letting 𝑦∈𝐾,then 𝑔(𝑦)∈𝐹1(⋅,𝑦(⋅))󳨐⇒𝑔(𝑦)∈𝑉󳨐⇒𝑁(𝑦) =𝐿(𝑔(𝑦))∈𝐾. (50) Now, we prove that 𝑁is continuous. Indeed, let 𝑦𝑛∈𝐾 converge to 𝑦in 𝐶([0,𝑏],R𝑁). Then 𝑔(𝑦𝑛)converge weakly to 𝑔(𝑦) as 𝑛󳨀→∞. (51) Since 𝑁(𝑦𝑛)=𝐿(𝑔(𝑦𝑛))∈𝐾and 𝑔(𝑦𝑛)(⋅)∈𝐹(𝑡,𝑦𝑛(𝑡)),then 𝑔(𝑦𝑛)(⋅)∈𝐹(⋅,𝐵𝑀)∈Pcp (R𝑁). (52) From Lemma 10,𝑔(𝑦𝑛)converge weakly to 𝑦in 𝐿1([0,𝑏],R𝑁) as 𝑛→∞. By the definition of 𝑁,wehave 𝑁(𝑦𝑛)=𝑦0+𝑦1𝑡+ 1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑔(𝑦𝑛)(𝑠)𝑑𝑠, 𝑡∈[0,𝑏], 𝑁(𝑦)=𝑦0+𝑦1𝑡+ 1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑔(𝑦)(𝑠)𝑑𝑠, 𝑡∈[0,𝑏]. (53) Since {𝑁(𝑦𝑛):𝑛∈N}⊂𝐾, then there exists subsequence of 𝑁(𝑦𝑛)converge in 𝐶([0,𝑏],R𝑁).Then 𝑁(𝑦𝑛)(𝑡)󳨀→ 𝑁(𝑦)(𝑡),∀𝑡∈ [0,𝑏],as 𝑛󳨀→∞.(54) Thisprovesthat𝑁is continuous. Hence by Schauder’s fixed point there exists 𝑦∈𝐾such that 𝑦=𝑁(𝑦). 5. The Relaxed Problem In this section, we examine whether the solutions of the extremal problem are dense in those of the convexified one. Such a result is important in optimal control theory whether the relaxed optimal state can be approximated by original states; the relaxed problems are generally much simpler to build. For the problem for first-order differential inclusions, we refer, for example, to [35,Theorem2,page124]or[36, Theorem 10.4.4, page 402]. For the relaxation of extremal problems we see the following recent references [30,50]. Now we present our main result of this section. Theorem 25. Let 𝐹:[0,𝑏]×R𝑁→P(R𝑁)be a multifunction satisfying the following hypotheses. (H3)The function 𝐹:[0,𝑏]×R𝑁→P𝑐𝑝𝑐V(R𝑁)such that, for all 𝑥∈R𝑁,themap 𝑡󳨃󳨀→ 𝐹(𝑡,𝑥)(55) is measurable. (H4)There exists 𝑝∈𝐿1(𝐽,R+)such that 𝐻𝑑(𝐹(𝑡,𝑥),𝐹(𝑡,𝑦))≤𝑝(𝑡)󵄩󵄩󵄩󵄩𝑥−𝑦󵄩󵄩󵄩󵄩, for a.e.𝑡∈[0,𝑏]and each 𝑥,𝑦∈R𝑁, 𝐻𝑑(𝐹(𝑡,0),0)≤𝑝(𝑡)for a.e.𝑡∈[0,𝑏]. (56) Then 𝑆𝑒=𝑆𝑐. Proof. By Coviz and Nadlar fixed point theorem, we can easily prove that 𝑆𝑐=0, and since 𝐹has compact and convex valued, then 𝑆𝑐is compact in 𝐶([0,𝑏],R𝑁).Formoreinformation we see [25,27–29,51,52]. Let 𝑦∈𝑆𝑐; then there exists 𝑓∈𝑆𝐹,𝑦 such that 𝑦(𝑡)=𝑦0+𝑦1𝑡+ 1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑓(𝑠)𝑑𝑠, a.e.𝑡∈[0,𝑏].(57) Let 𝐾be a compact and convex set in 𝐶([0,𝑏],R𝑁)such that 𝑆𝑐⊂𝐾.Giventhat𝑦∗∈𝐾and 𝜖>0, we define the following multifunction 𝑈𝜖:[0,𝑏]→P(R𝑁)by 𝑈𝜖(𝑡)={𝑢∈R𝑁:󵄩󵄩󵄩󵄩𝑓(𝑡)−𝑢󵄩󵄩󵄩󵄩<𝑑(𝑓(𝑡),𝐹(𝑡,𝑦(𝑡)))+𝜖, 𝑢∈𝐹(𝑡,𝑦∗(𝑡))}. (58) Abstract and Applied Analysis 7 The multivalued map 𝑡→𝐹(𝑡,⋅)is measurable and 𝑥→ 𝐹(⋅,𝑥)is 𝐻𝑑continuous. In addition, if 𝐹(⋅,⋅)has compact values, then 𝐹(⋅,⋅) is graph measurable, and the mapping 𝑡 → 𝐹(𝑡,𝑦(𝑡))is a measurable multivalued map for fixed 𝑦∈ 𝐶([0,𝑏],R𝑁).ThenbyLemma 3, there exists a measurable selection V1(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e.𝑡∈[0,𝑏]such that 󵄩󵄩󵄩󵄩𝑓(𝑡)−V1(𝑡)󵄩󵄩󵄩󵄩<𝑑(𝑓(𝑡),𝐹(𝑡,𝑦(𝑡)))+𝜖; (59) this implies that 𝑈𝜖(⋅) =0. We consider 𝐺𝜖:𝐾→P(𝐿1(𝐽, R𝑁)defined by 𝐺𝜖(𝑦)={𝑓∗∈F(𝑦):󵄩󵄩󵄩󵄩𝑓(𝑡)−𝑓∗(𝑡)󵄩󵄩󵄩󵄩 <𝜖+𝑑(𝑓∗(𝑡),𝐹(𝑡,𝑦(𝑡)))}. (60) Since the measurable multifunction 𝐹is integrable bounded, Lemma 9 implies that the Nemyts’ki˘ ıoperatorFhas decomposable values. Hence 𝑦→𝐺𝜖(𝑦)is l.s.c. with decomposable values. By Lemma 8, there exists a continuous selection 𝑓𝜖: 𝐶([0,𝑏],R𝑁)→𝐿 1(𝐽,R𝑁)such that 𝑓𝜖(𝑦)∈𝐺𝜖(𝑦) ∀𝑦∈𝐶([0,𝑏],R𝑁).(61) From Theorem 17, there exists function 𝑔𝜖:𝐾→𝐿𝑤([0,𝑏], R𝑁)such that 𝑔𝜖(𝑦)∈ext 𝑆𝐹(𝑦)=𝑆ext 𝐹(⋅,𝑦(⋅)) ∀𝑦∈𝐾, ‖𝑔𝜖(𝑦)−𝑓𝜖(𝑦)‖𝑤≤𝜖, ∀𝑦∈𝐾. (62) From (H3)we can prove that there exists 𝑀>0such that 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩∞≤𝑀 ∀𝑦∈𝑆𝑐.(63) Consider the sequence 𝜖𝑛→0,as𝑛→∞,andset𝑔𝑛=𝑔𝜖𝑛, 𝑓𝑛=𝑓𝜖𝑛.Set 𝑉={𝑓∈𝐿1([0,𝑏],R𝑁):󵄩󵄩󵄩󵄩𝑓(𝑡)󵄩󵄩󵄩󵄩≤𝜓(𝑡)a.e.𝑡∈[0,𝑏]}, 𝜓(𝑡)=(1+𝑀)𝑝(𝑡). (64) Let 𝐿:𝑉 → 𝐶([0,𝑏],R𝑁)be the map such that each 𝑓∈𝑉 assigns the unique solution of the problem 𝑐𝐷𝛼𝑦(𝑡)=𝑓(𝑡),a.e.𝑡∈[0,𝑏], 𝑦(0)=𝑦0,𝑦 󸀠(0)=𝑦1.(65) As in Theorem 24,wecanprovethat𝐿(𝑉)is compact in 𝐶([0,𝑏],R𝑁)and the operator 𝑁𝑛=𝐿∘𝑔 𝑛:𝐾→𝐾is compact; then by Schauder’s fixed point there exists  𝑦𝑛∈𝐾 such that  𝑦𝑛∈𝑆𝑒and  𝑦𝑛(𝑡)=𝑦0+𝑡𝑦1+1 Γ(𝛼)∫𝑡 0(𝑡−𝑠)𝛼−1𝑔𝑛(𝑦𝑛)(𝑠)𝑑𝑠, a.e.𝑡∈[0,𝑏],𝑛∈N.(66) Hence 󵄩󵄩󵄩󵄩𝑦(𝑡)− 𝑦𝑛(𝑡)󵄩󵄩󵄩󵄩 ≤1 Γ(𝛼)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0(𝑡−𝑠)𝛼−1 [𝑔𝑛( 𝑦𝑛)(𝑠)−𝑓(𝑠)]𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤1 Γ(𝛼)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0(𝑡−𝑠)𝛼−1 [𝑔𝑛( 𝑦𝑛)(𝑠)−𝑓𝑛( 𝑦𝑛)(𝑠)]𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 +𝑏𝛼 Γ(𝛼)∫𝑡 0󵄩󵄩󵄩󵄩𝑓𝑛( 𝑦𝑛)(𝑠)−𝑓(𝑠)󵄩󵄩󵄩󵄩𝑑𝑠 ≤1 Γ(𝛼)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0(𝑡−𝑠)𝛼−1 [𝑔𝑛( 𝑦𝑛)(𝑠)−𝑓𝑛( 𝑦𝑛)(𝑠)]𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 +𝑏𝛼 Γ(𝛼)∫𝑡 0(𝜖𝑛+𝑑(𝑓(𝑠),𝑓𝑛( 𝑦𝑛)(𝑠)))𝑑𝑠 ≤1 Γ(𝛼)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫𝑡 0(𝑡−𝑠)𝛼−1 [𝑔𝑛( 𝑦𝑛)(𝑠)−𝑓𝑛( 𝑦𝑛)(𝑠)]𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 +𝑏𝛼 Γ(𝛼)∫𝑡 0(𝜖𝑛+𝐻𝑑(𝐹(𝑠,𝑦(𝑠)),𝐹(𝑠, 𝑦𝑛(𝑠))))𝑑𝑠 ≤𝑏𝛼+1 Γ(𝛼+1)𝜖𝑛+𝑏𝛼+1 Γ(𝛼)𝜖𝑛+∫𝑡 0𝑝(𝑠)󵄩󵄩󵄩󵄩𝑦(𝑠)− 𝑦𝑛(𝑠)󵄩󵄩󵄩󵄩.(67) Let  𝑦(⋅)be a limit point of the sequence  𝑦𝑛(⋅).Then,itfollows that from the above inequality, one has 󵄩󵄩󵄩󵄩𝑦(𝑡)− 𝑦(𝑡)󵄩󵄩󵄩󵄩≤∫𝑡 0𝑝(𝑠)󵄩󵄩󵄩󵄩𝑦(𝑠)− 𝑦(𝑠)󵄩󵄩󵄩󵄩𝑑𝑠, (68) which implies 𝑦(⋅)=  𝑦(⋅).Consequently,𝑦∈𝑆𝑐is a unique limit point of  𝑦𝑛(⋅)∈𝑆𝑒. Example 26. Let 𝐹:𝐽×R𝑁→Pcpcv(R𝑁)with 𝐹(𝑡,𝑦)=𝐵(𝑓1(𝑡,𝑦),𝑓2(𝑡,𝑦)), (69) where 𝑓1,𝑓2:𝐽×R𝑁→R𝑁are Carath´ eodory functions and bounded. Then (2)issolvable. Example 27. If, in addition to the conditions on 𝐹of Example 26,𝑓1and 𝑓2are Lipschitz functions, then 𝑆𝑒=𝑆𝑐. Acknowledgments This work is partially supported by the Ministerio de Economia y Competitividad, Spain, project MTM2010-15314, and cofinanced by the European Community Fund FEDER. 8Abstract and Applied Analysis References [1]K.DiethelmandA.D.Freed,“Onthesolutionofnonlinear fractional order differential equations used in the modeling of viscoplasticity,” in Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer, Heidelberg, Germany, 1999. [2] L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing,vol.5,no.2,pp.81–88,1991. [3] W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol.68,no.1,pp.46–53,1995. [4] F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics,A.CarpinteriandF. Mainardi, Eds., pp. 291–348, Springer, Wien, Germany, 1997. [5]A.B.MalinowskaandD.F.M.Torres,“Towardsacombined fractional mechanics and quantization,” Fractional Calculus and Applied Analysis,vol.15,no.3,pp.407–417,2012. [6] R. Metzler, W. Schick, H. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” TheJournalofChemicalPhysics,vol.103,no.16,pp.7180–7186, 1995. [7] S. P. N¨ asholm and S. Holm, “On a fractional Zener elastic wave equation,” Fractional Calculus and Applied Analysis,vol.16,no. 1,pp.26–50,2013. [8] L. V´ azquez, J. J. Trujillo, and M. P. Velasco, “Fractional heat equation and the second law of thermodynamics,” Fractional Calculus and Applied Analysis,vol.14,no.3,pp.334–342,2011. [9] B. J. West and D. West, “Fractional dynamics of allometry,” Fractional Calculus and Applied Analysis,vol.15,no.1,pp.70– 96, 2012. [10] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,vol.204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherland, 2006. [11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, NY, USA, 1993. [12] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. [13]S.G.Samko,A.A.Kilbas,andO.I.Marichev,Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. [14] A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis,vol.81,no.2,pp.435–493,2002. [15] A. M. Nahuˇ sev, “The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms,” Doklady Akademii Nauk SSSR,vol.234,no. 2, pp. 308–311, 1977. [16] I. Podlubny, I. Petr´ aˇ s, B. M. Vinagre, P. O’Leary, and L. Dorˇ c´ ak, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics,vol.29,no.1–4,pp.281–296,2002. [17] C. Yu and G. Gao, “Existence of fractional differential equations,” Journal of Mathematical Analysis and Applications,vol. 310,no.1,pp.26–29,2005. [18] V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications A,vol.69,no.10,pp.3337–3343,2008. [19] Y. Chalco-Cano, J. J. Nieto, A. Ouahab, and H. Rom´ an-Flores, “Solution set for fractional differential equations with Riemann- Liouville derivative,” Fractional Calculus and Applied Analysis, vol.16,no.3,pp.682–694,2013. [20] J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation,vol.16,no.3,pp.1140–1153,2011. [21] S. Abbas and M. Benchohra, “Fractional order Riemann- Liouville integral inclusions with two independent variables and multiple delay,” Opuscula Mathematica,vol.33,no.2,pp. 209–222, 2013. [22] S. Abbas, M. Benchohra, and J. J. Nieto, “Global uniqueness results for fractional order partial hyperbolic functional differential equations,” Advances in Difference Equations, vol. 2011, Article ID 379876, 25 pages, 2011. [23] M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional functional differential inclusions with infinite delay and applications to control theory,” Fractional Calculus & Applied Analysis,vol.11,no.1,pp.35–56, 2008. [24] J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” Nonlinear Analysis: Theory, Methods & Applications A,vol.70,no.5,pp.2091–2105,2009. [25] J. Henderson and A. Ouahab, “Impulsive differential inclusions with fractional order,” Computers & Mathematics with Applications,vol.59,no.3,pp.1191–1226,2010. [26] F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications,vol.62,no.3,pp. 1181–1199, 2011. [27] A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear Analysis: Theory, Methods & Applications A,vol.69,no.11,pp.3877–3896,2008. [28] A. Ouahab, “Filippov’s theorem for impulsive differential inclusions with fractional order,” Electronic Journal of Qualitative Theory of Differential Equations,no.23,pp.1–23,2009. [29] A. Ouahab, “Fractional semilinear differential inclusions,” Computers & Mathematics with Applications,vol.64,no.10,pp. 3235–3252, 2012. [30] F. S. de Blasi and G. Pianigiani, “Non-convex valued differential inclusions in Banach spaces,” Journal of Mathematical Analysis and Applications,vol.128,pp.541–555,1996. [31] N. S. Papageorgiou, “On the “bang-bang” principle for nonlinear evolution inclusions,” Aequationes Mathematicae,vol.45, no. 2-3, pp. 267–280, 1993. [32] A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic, Dordrecht, The Netherlands, 2000. [33] A. A. Tolstonogov, “Extreme continuous selectors of multivalued maps and their applications,” Journal of Differential Equations,vol.122,no.2,pp.161–180,1995. [34] A. A. Tolstonogov, “Extremal selectors of multivalued mappings and the, “bangbang” principle for evolution inclusions,” Doklady Akademii Nauk SSSR,vol.317,no.3,pp.589–593,1991 (Russian), translation in Soviet Mathematics Doklady,vol.43, no. 2, 481–485, 1991. [35] J.-P. Aubin and A. Cellina, Differential Inclusions,Springer, Berlin, Germany, 1984. Abstract and Applied Analysis 9 [36] J.-P. Aubin and H. Frankowska, Set-Valued Analysis,Birkh ¨ auser, Boston, Mass, USA, 1990. [37] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis,VolumeI:Theory, Kluwer Academic Publishers, London, UK, 1997. [38] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic, Dordrecht, The Netherlands, 1991. [39] Q. J. Zhu, “On the solution set of differential inclusions in Banach space,” Journal of Differential Equations,vol.93,no.2, pp. 213–237, 1991. [40] A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica,vol.90,no.1, pp. 70–85, 1988. [41] M. Frigon and A. Granas, “Th´ eor` emes d’existence pour des inclusions diff´ erentielles sans convexit´ e,” ComptesRendusde l’Acad´ emie des Sciences. S´ erie I,vol.310,no.12,pp.819–822,1990. [42] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, With the Assistance of W. G. Bade and R. G. Bartle, Pure and Applied Mathematics, Interscience Publishers, New York, NY, USA, 1958. [43] R. D. Bourgin, Geometric Aspects of Convex Sets with the Radon- Nikody’m Property,vol.993ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1983. [44] M. Florenzano and C. Le Van, Finite Dimensional Convexity and Optimization,vol.13ofStudies in Economic Theory,Springer, Berlin, Germany, 2001, In Cooperation with Pascal Gourdel. [45] S. Abbas, M. Benchohra, and G. M. N’Gu´ er´ ekata, Topics in Fractional Differential Equations,vol.27ofDevelopments in Mathematics, Springer, New York, NY, USA, 2012. [46] M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969. [47] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529–529, 1967. [48] M. Caputo and F. Mainardi, “Linear models of dissipation in anelastic solids,” La Rivista del Nuovo Cimento,vol.1,no.2,pp. 161–198, 1971. [49] M. Benamara, Point extremaux, multi-applications et fonctionnelles integrales [Th´ ese de 3´ eme Cycle],UniversitedeGrenoble, Grenoble, France, 1975. [50] F. S. de Blasi and G. Pianigiani, “Baire’s category and the bangbang property for evolution differential inclusions of contractive type,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 550–567, 2010. [51] B. Ahmad and J. J. Nieto, “A study of impulsive fractional differential inclusions with anti-periodic boundary conditions,” Fractional Differential Calculus,vol.2,no.1,pp.1–15,2012. [52] B. Ahmad and J. J. Nieto, “Existence results for higher order fractional differential inclusions with nonlocal boundary conditions,” Nonlinear Studies,vol.17,no.2,pp.131–138,2010.