Extremal Solutions and Relaxation Problems for Fractional Differential Inclusions

Abstract

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] and the papers of Kilbas and Trujillo [14], Nahušev [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] and Chalco-Cano et al. [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] and in the 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: ^𝑐𝐷^𝛼𝑦(𝑡)(𝑡[],𝑦(0)∈𝐹𝑡,𝑦),a.e.𝑡∈𝐽∶=0,𝑏=𝑦₀,𝑦^′(0)=𝑦₁,_𝑐𝐷^𝛼𝑦(𝑡)(𝑡)[],𝑦(0)∈ext𝐹𝑡,𝑦,a.e.𝑡∈𝐽∶=0,𝑏=𝑦₀,𝑦^′(0)=𝑦₁, (1)(2) where ^𝑐𝐷^𝛼 is the Caputo fractional derivatives, 𝛼∈(1,2], 𝐹∶𝐽×ℝ^𝑁→𝒫(ℝ^𝑁) is a multifunction, and ext𝐹(𝑡,𝑦) represents the set of extreme points of 𝐹(𝑡,𝑦). (𝒫(ℝ^𝑁) is the family of all nonempty subsets of ℝ^𝑁.

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.

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 Sections 2 and 3, respectively. Then, we will be concerned with the existence of solution for extremal problem. This is the aim of Section 4. In Section 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) the Banach space of all continuous functions from 𝐽 into 𝑋 with the norm 𝑦_∞=sup𝑦(𝑡)∶0≤𝑡≤𝑏. (3)

A measurable function 𝑦∶[0,𝑏]→𝑋 is 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 𝑦₁=^𝑏₀𝑦(𝑡)𝑑𝑡. (4) Denote by 𝐿¹_𝑤([0,𝑏],𝑋) the space of equivalence classes of Bochner integrable function 𝑦∶[0,𝑏]→𝑋 with the norm 𝑦_𝑤=sup_{𝑡∈[0,𝑡]}^𝑡₀𝑦(𝑠)𝑑𝑠. (5) The norm ‖⋅‖_𝑤 is weaker than the usual norm ‖⋅‖₁, and for a broad class of subsets of 𝐿¹([0,𝑏],𝑋), the topology defined by the weak norm coincides with the usual weak topology (see [37, Proposition 4.14, page 195]). Denote by𝒫(𝑋)={},𝒫𝑌⊂𝑋∶𝑌≠∅_cl(𝑋)={(𝑋)},𝒫𝑌∈𝒫∶𝑌closed_𝑏(𝑋)={(𝑋)},𝒫𝑌∈𝒫∶𝑌bounded_cv(𝑋)={(𝑋)},𝒫𝑌∈𝒫∶𝑌convex_cp(𝑋)=𝑌∈𝒫(𝑋)∶𝑌compact. (6) A multivalued map 𝐺∶𝑋→𝒫(𝑋) has convex (closed) values if 𝐺(𝑥) is convex (closed) for all 𝑥∈𝑋. We say that 𝐺 is bounded on bounded sets if 𝐺(𝐵) is bounded in 𝑋 for each bounded set 𝐵 of 𝑋 (i.e., sup_𝑥∈𝐵{sup{‖𝑦‖∶𝑦∈𝐺(𝑥)}}<∞).

Definition 1. A multifunction 𝐹∶𝑋→𝒫(𝑌) is said to be upper semicontinuous at the point 𝑥₀∈𝑋, if, for every open 𝑊⊆𝑌 such that 𝐹(𝑥₀)⊂𝑊, there exists a neighborhood 𝑉(𝑥₀) of 𝑥₀ such that 𝐹(𝑉(𝑥₀))⊂𝑊.

A multifunction is called upper semicontinuous (u.s.c. for short) on 𝑋 if for each 𝑥∈𝑋 it 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, for every open 𝑊⊆𝑌 such that 𝐹(𝑥₀)∩𝑊≠∅, there exists a neighborhood 𝑉(𝑥₀) of 𝑥₀ with property that 𝐹(𝑥)∩𝑊≠∅ for 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]→𝑌 a measurable function. Then for any measurable 𝑣∶[𝑎,𝑏]→(0,+∞), there exists a measurable selection 𝑓_𝑣 of 𝐹 such that for a.e. 𝑡∈[𝑎,𝑏], 𝑢(𝑡)−𝑓_𝑣(𝑡)≤𝑑(𝑢(𝑡),𝐹(𝑡))+𝑣(𝑡). (7)

First, consider the Hausdorff pseudometric 𝐻_𝑑∶𝒫(𝐸)×𝒫(𝐸)⟶ℝ⁺∪{∞}, (8) defined by 𝐻_𝑑(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝐴,𝑏), (9) where 𝑑(𝐴,𝑏)=inf_𝑎∈𝐴𝑑(𝑎,𝑏) and 𝑑(𝑎,𝐵)=inf_𝑏∈𝐵𝑑(𝑎,𝑏). (𝒫_𝑏,cl(𝐸),𝐻_𝑑) is a metric space and (𝒫_cl(𝑋),𝐻_𝑑) is a generalized metric space.

Definition 4. A multifunction 𝐹∶𝑌→𝒫(𝑋) is called Hausdorff lower semicontinuous at the point 𝑦₀∈𝑌, if for any 𝜖>0 there exists a neighbourhood 𝑈(𝑦₀) of the point 𝑦₀ such that 𝐹𝑦₀⊂𝐹𝑦+𝜖𝐵(0,1),forevery𝑦∈𝑈𝑦₀, (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, if for any 𝜖>0 there exists a neighbourhood 𝑈(𝑦₀) of the point 𝑦₀ such that 𝐹𝑦⊂𝐹𝑦₀+𝜖𝐵(0,1),forevery𝑦∈𝑈𝑦₀. (11)

𝐹 is called continuous, if it is Hausdorff lower and upper semicontinuous.

Definition 6. Let 𝑋 be a Banach space; a subset 𝐴⊂𝐿¹([0,𝑏],𝑋) is decomposable if, for all 𝑢,𝑣∈𝐴 and for every Lebesgue measurable set 𝐼⊂𝐽, one has 𝑢𝜒_𝐼+𝑣𝜒_{[0,𝑏]⧵𝐼}∈𝐴, (12) where 𝜒_𝐴 stands for the characteristic function of the set 𝐴. We denote by Dco(𝐿¹([0,𝑏],𝑋)) the family of decomposable sets.

Let 𝐹∶[0,𝑏]×𝑋→𝒫(𝑋) be a multivalued map with nonempty closed values. Assign to 𝐹 the multivalued operator ℱ∶𝐶([0,𝑏],𝑋)→𝒫(𝐿¹([0,𝑏],𝑋)) defined by ℱ𝑦=𝑣∈𝐿¹])(𝑡)(𝑡),[.([0,𝑏,𝑋∶𝑣∈𝐹𝑡,𝑦a.e.𝑡∈0,𝑏] (13) The operator ℱ is called the Nemyts’kiĭ operator associated to 𝐹.

Definition 7. Let 𝐹∶[0,𝑏]×𝑋→𝒫(𝑋) be a multivalued map with nonempty compact values. We say that 𝐹 is of lower semicontinuous type (l.s.c. type) if its associated Nemyts’kiĭ operator ℱ is 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 𝑋 be a separable metric space and let 𝐸 be a Banach space. Then every l.s.c. multivalued operator 𝑁∶𝑋→𝒫_𝑐𝑙(𝐿¹([0,𝑏],𝐸)) with closed decomposable values has a continuous selection; that is, there exists a continuous single-valued 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→𝒫_𝑐𝑝(𝐸) be an integrably bounded multivalued map satisfying (ℋ₁). Then 𝐹 is of lower semicontinuous type.

Define 𝐹(𝐾)=𝑓∈𝐿¹([0,𝑏],𝑋)∶𝑓(𝑡)∈𝐾a.e.𝑡∈[0,𝑏],𝐾⊂𝑋, (14) where 𝑋 is a Banach space.

Lemma 10 (see [37]). Let 𝐾⊂𝑋 be a weakly compact subset of 𝑋. Then 𝐹(𝐾) is relatively weakly compact subset of 𝐿¹([0,𝑏],𝑋). Moreover if 𝐾 is convex, then 𝐹(𝐾) is weakly compact in 𝐿¹([0,𝑏],𝑋).

Definition 11. A multifunction 𝐹∶[0,𝑏]×𝑌→𝒫_𝑤cpcv(𝑋) possesses the Scorza-Dragoni property (S-D property) if for each 𝜖>0, there exists a closed set 𝐽_𝜖⊂[0,𝑏] whose Lebesgue measure 𝜇(𝐽_𝜖)≤𝜖 and such that 𝐹∶[0,𝑏]⧵𝐽_𝜖×𝑌→𝑋 is continuous with respect to the metric 𝑑_𝑋(⋅,⋅).

Remark 12. It is well known that if the map 𝐹∶[0,𝑏]×𝑌→𝒫_𝑤cpcv(𝑋) is continuous with respect to 𝑦 for almost every 𝑡∈[0,𝑏] and is measurable with respect to 𝑡 for 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 𝐴 be a nonempty subset of a real or complex linear vector space. An extreme point of a convex set 𝐴 is a point 𝑥∈𝐴 with the property that if 𝑥=𝜆𝑦+(1−𝜆)𝑧 with 𝑦,𝑧∈𝐴 and 𝜆∈[0,1], then 𝑦=𝑥 and/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∈ℕ be a denumerable, dense (in 𝜎(𝑋^′,𝑋) topology) subset of the set {𝑥∈𝑋∶‖𝑥‖≤1}. For any 𝐴∈𝒫_cpcv(𝑋) and 𝑥^′_𝑛 define the function 𝑑^𝑛(𝐴,𝑢)=max𝑦−𝑧,𝑥^′_𝑛∶𝑦,𝑧∈𝐴,𝑢=𝑦+𝑧2. (15)

Lemma 15 (see [33]). 𝑢∈ext(𝐴) if and only if 𝑑^𝑛(𝐴,𝑢)=0 for all 𝑛≥1.

In accordance with Krein-Milman and Trojansky theorem [43], the set ext(𝑆_𝐹) is nonempty and co(ext(𝑆_𝐹))=𝑆_𝐹.

Lemma 16 (see [33]). Let 𝐹∶[0,𝑏]→𝒫_{𝑤𝑐𝑝𝑐𝑣}(𝑋) be a measurable, integrably bounded map. Then ext𝑆_𝐹⊆𝑆_𝐹, (16) where ext(𝑆_𝐹) is the closure of set ext(𝑆_𝐹) in the topology of the space 𝐿¹([0,𝑏],𝑋).

Theorem 17 (see [33]). Let 𝐹∶[0,𝑏]×𝑌→𝒫_{𝑤𝑐𝑝𝑐𝑣}(𝑋) be a multivalued map that has the 𝑆-𝐷 property and let it be integrable bounded on compacts from 𝑌. Consider a compact subset 𝐾⊂𝐶([0,𝑏],𝑋) and define the multivalued map 𝐺∶𝐾→𝐿¹([0,𝑏],𝑋), by 𝐺𝑦(⋅)=𝑓∈𝐿¹([0,𝑏],𝑋)∶𝑓(𝑡)∈𝐹𝑡,𝑦(𝑡)𝑎.𝑒.𝑜𝑛[0,𝑏],𝑦∈𝐾. (17) Then for every 𝐾 compact in 𝐶([0,𝑏],𝑋),  𝜖>0 and any continuous selection 𝑓∶𝐾→𝐿¹([0,𝑏],𝑋), there exists a continuous selector 𝑔∶𝐾→𝐿¹([0,𝑏],𝑋) of the map ext(𝐺)∶𝐾→𝐿¹([0,𝑏],𝑋) such that for all 𝑦∈𝐶([0,𝑏],𝑋) one has sup_{𝑡∈[0,𝑏]}^𝑡₀𝑓𝑦(𝑠)−𝑔𝑦(𝑠)𝑑𝑠≤𝜖. (18)

For a background of extreme point of 𝐹(𝑡,𝑦(𝑡)) see Dunford-Schwartz [42, Chapter 5, Section 8] and Florenzano and Le Van [44, Chapter 3].

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 𝐼^𝑛𝑓(𝑡)1∶=(𝑛−1)!^𝑡₀(𝑡−𝑠)^𝑛−1𝑓(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (19)

Definition 18 (see [13, 45]). The fractional integral of order 𝛼>0 of a function 𝑓∈𝐿¹([𝑎,𝑏],ℝ) is defined by 𝐼^𝛼_𝑎+𝑓(𝑡)=^𝑡_𝑎(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠)𝑑𝑠, (20) where Γ is the gamma function. When 𝑎=0, we write 𝐼^𝛼𝑓(𝑡)=𝑓(𝑡)∗𝜙_𝛼(𝑡), where 𝜙_𝛼(𝑡)=𝑡^(𝛼−1)/Γ(𝛼) for 𝑡>0, and we write 𝜙_𝛼(𝑡)=0 for 𝑡≤0 and 𝜙_𝛼→𝛿(𝑡) as 𝛼→0, where 𝛿 is the delta function and Γ is the Euler gamma function defined by Γ(𝛼)=^∞₀𝑡^𝛼−1𝑒^−𝑡𝑑𝑡,𝛼>0. (21) For consistency, 𝐼⁰=Id (identity operator), that is, 𝐼⁰𝑓(𝑡)=𝑓(𝑡). Furthermore, by 𝐼^𝛼𝑓(0⁺) we mean the limit (if it exists) of 𝐼^𝛼𝑓(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 𝛼 with −𝛼 in 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 𝐷^𝑛, with 𝑛∈ℕ, the operator of the derivative of order 𝑛, we first note that 𝐷^𝑛𝐼^𝑛=Id,𝐼^𝑛𝐷^𝑛≠Id,𝑛∈ℕ, (22) that is, 𝐷^𝑛 is the left inverse (and not the right inverse) to the corresponding integral operator 𝐽^𝑛. We can easily prove that 𝐼^𝑛𝐷^𝑛𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)𝑎⁺(𝑡−𝑎)^𝑘𝑘!,𝑡>0. (23) As a consequence, we expect that 𝐷^𝛼 is defined as the left inverse to 𝐼^𝛼. For this purpose, introducing the positive integer 𝑛 such that 𝑛−1<𝛼≤𝑛, one defines the fractional derivative of order 𝛼>0.

Definition 19. For a function 𝑓 given on interval [𝑎,𝑏], the 𝛼th Riemann-Liouville fractional-order derivative of 𝑓 is defined by 𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡_𝑎(𝑡−𝑠)^{−𝛼+𝑛−1}𝑓(𝑠)𝑑𝑠, (24) where 𝑛=[𝛼]+1 and [𝛼] is the integer part of 𝛼.

Defining for consistency, 𝐷⁰=𝐼⁰=Id, then we easily recognize that 𝐷^𝛼𝐼^𝛼𝐷=Id,𝛼≥0,_𝛼𝑡^𝛾=Γ𝛾+1Γ𝑡𝛾+1−𝛼^𝛾−𝛼,𝛼>0,𝛾∈(−1,0)∪(0,+∞),𝑡>0. (25)(26) Of course, properties (25) and (26) are a natural generalization of those known when the order is a positive integer.

Note the remarkable fact that the fractional derivative 𝐷^𝛼𝑓 is not zero for the constant function 𝑓(𝑡)=1, if 𝛼∉ℕ. In fact, (26) with 𝛾=0 illustrates that 𝐷^𝛼()1=𝑡−𝑎^−𝛼Γ(1−𝛼),𝛼>0,𝑡>0. (27) It is clear that 𝐷^𝛼1=0, for 𝛼∈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] in the 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 fractional-order derivative of 𝑓 is defined by ^𝑐𝐷^𝛼𝑓𝑡)1(∶=Γ(𝑛−𝛼)^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^𝑛(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. Whenever we use the operator 𝐷^𝛼_∗ we (tacitly) assume that this condition is met. We easily recognize that in general 𝐷^𝛼𝑓(𝑡)∶=𝐷^𝑚𝐼^𝑚−𝛼𝑓(𝑡)≠𝐽^𝑚−𝛼𝐷^𝑚𝑓(𝑡)∶=𝐷^𝛼_∗𝑓(𝑡), (29) unless the function 𝑓(𝑡), along with its first 𝑛−1 derivatives, vanishes at 𝑡=𝑎⁺. In fact, assuming that the passage of the 𝑚-derivative under the integral is legitimate, we recognize that, for 𝑚−1<𝛼<𝑚 and 𝑡>0, 𝐷^𝛼𝑓(𝑡)=^𝑐𝐷^𝛼𝑓(𝑡)+^𝑚−1_𝑘=0(𝑡−𝑎)^𝑘−𝛼Γ()𝑓𝑘−𝛼+1^(𝑘)𝑎⁺, (30) and therefore, recalling the fractional derivative of the power function (26), one has 𝐷^𝛼𝑓(𝑡)−^𝑚−1_𝑘=0(𝑡−𝑎)^𝑘−𝛼Γ()𝑓𝑘−𝛼+1^(𝑘)𝑎⁺=𝐷^𝛼_∗𝑓(𝑡). (31) The alternative definition, that is, Definition 20, for the fractional derivative thus incorporates the initial values of the function and of lower order. The subtraction of the Taylor polynomial of degree 𝑛−1 at 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:^𝑐𝐷^𝛼1=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 𝐷^𝛼(𝑡−𝑎)^𝛼−𝑗=0,for𝑗=1,2,…,[𝛼]+1. (33) From (32) and (33) we thus recognize the following statements about functions which, for 𝑡>0, admit the same fractional derivative of order 𝛼, with 𝑛−1<𝛼≤𝑛,  𝑛∈ℕ: 𝐷^𝛼𝑓(𝑡)=𝐷^𝛼𝑔(𝑡)⟺𝑓(𝑡)=𝑔(𝑡)+^𝑚_𝑗=1𝑐_𝑗(𝑡−𝑎)^𝛼−𝑗,_𝑐𝐷^𝛼𝑓(𝑡)=^𝑐𝐷^𝛼𝑔(𝑡)⟺𝑓(𝑡)=𝑔(𝑡)+^𝑚_𝑗=1𝑐_𝑗(𝑡−𝑎)^𝑛−𝑗. (34) In these formulas, the coefficients 𝑐_𝑗 are arbitrary constants. For proving all main results we present the following auxiliary lemmas.

Lemma 21 (see [10]). Let 𝛼>0 and let 𝑦∈𝐿^∞(𝑎,𝑏) or 𝐶([𝑎,𝑏]). Then ^𝑐𝐷^𝛼𝐼^𝛼𝑦(𝑡)=𝑦(𝑡). (35)

Lemma 22 (see [10]). Let 𝛼>0 and 𝑛=[𝛼]+1. If 𝑦∈𝐴𝐶^𝑛[𝑎,𝑏] or 𝑦∈𝐶^𝑛[𝑎,𝑏], then 𝐼^𝛼𝑐𝐷^𝛼𝑦(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], and Caputo [46–48].

4. Existence Result

Definition 23. A function 𝑦∈𝐶([0,𝑏],ℝ^𝑁) is called mild solution of problem (1) if there exist 𝑓∈𝐿¹(𝐽,ℝ^𝑁) such that 𝑦(𝑡)=𝑦₀+𝑡𝑦₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^1−𝛼𝑓(𝑠)𝑑𝑠,𝑡∈[0,𝑏], (37) where 𝑓∈𝑆_𝐹,𝑦={𝑣∈𝐿¹([0,𝑏],ℝ^𝑁)∶𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e.on[0,𝑏]}.

We will impose the following conditions on 𝐹.(ℋ₁)The function 𝐹∶𝐽×ℝ^𝑁→𝒫_cpcv(ℝ^𝑁) such that(a)for all 𝑥∈ℝ^𝑁, the map 𝑡↦𝐹(𝑡,𝑥) is measurable,(b)for every 𝑡∈[0,𝑏], the multivalued map 𝑥→𝐹(𝑡,𝑥) is 𝐻_𝑑 continuous (ℋ₂)There exist 𝑝∈𝐿¹(𝐽,ℝ⁺) and a continuous nondecreasing function 𝜓∶[0,∞)→(0,∞) such that ‖𝐹(𝑡,𝑥)‖_𝒫𝑣‖((𝑡)𝜓𝑥,[]=sup{‖∶𝑣∈𝐹𝑡,𝑥)}≤𝑝(‖‖)fora.e.𝑡∈0,𝑏andeach𝑥∈ℝ^𝑁, (38) with ^𝑏₀𝑝(𝑠)𝑑𝑠<^∞_{‖𝑦0‖+𝑏‖𝑦1‖}𝑑𝑢𝜓(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>0 such that ‖𝑦‖_∞≤𝑀 for each 𝑦∈𝑆_𝑐.
Let 𝐹₁=𝐹𝑦𝐹𝑡,𝑦𝑡,𝑦if≤𝑀,𝑡,𝑀𝑦𝑦if𝑦≥𝑀. (40) We consider ^𝑐𝐷^𝛼𝑦(𝑡)∈𝐹₁(𝑡)[],𝑦(0)𝑡,𝑦,a.e.𝑡∈0,𝑏=𝑦₀,𝑦^′(0)=𝑦₁. (41) It is clear that all the solutions of (41) are solutions of (2).
Set 𝑉=𝑓∈𝐿¹[0,𝑏],ℝ^𝑁∶𝑓(𝑡)≤𝜓_∗(𝑡,𝜓_∗)(𝑡)=𝑝(𝑡)𝜓(𝑀). (42) It is clear that 𝑉 is weakly compact in 𝐿¹([0,𝑏],ℝ^𝑁). Remark that for every 𝑓∈𝑉, there exists a unique solution 𝐿(𝑓) of the following problem: ^𝑐𝐷^𝛼𝑦(𝑡)(𝑡)[],𝑦(𝑡)=𝑓,a.e.𝑡∈0,𝑏=𝑦₀,𝑦^′(0)=𝑦₁; (43) this solution is defined by 𝐿𝑓(𝑡)=𝑦₀+𝑡𝑦₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)[].𝑑𝑠,a.e.𝑡∈0,𝑏 (44) We claim that 𝐿 is continuous. Indeed, let 𝑓_𝑛→𝑓 converge in 𝐿¹([0,𝑏],ℝ^𝑁), as 𝑛→∞, set 𝑦_𝑛=𝐿(𝑓_𝑛), 𝑛∈ℕ. It is clear that {𝑦_𝑛∶𝑛∈ℕ} is relatively compact in 𝐶([0,𝑏],ℝ^𝑁) and 𝑦_𝑛 converge to 𝑦∈𝐶([0,𝑏],ℝ^𝑁). Let 𝑧(𝑡)=𝑦₀+𝑦₁1𝑡+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡∈[0,𝑏]. (45) Then 𝑦_𝑛−𝑧_∞≤𝑏^𝛼Γ(𝛼)^𝑏₀𝑓_𝑛(𝑠)−𝑓(𝑠)𝑑𝑠⟶0,as𝑛⟶∞. (46) Hence 𝐾=𝐿(𝑉) is compact and convex subset of 𝐶([0,𝑏],ℝ^𝑁). Let 𝑆_𝐹∶𝐾→𝒫_clcv(𝐿¹([0,𝑏],ℝ^𝑁)) be the multivalued Nemitsky operator defined by 𝑆_𝐹1𝑦=𝑓∈𝐿¹[0,𝑏],ℝ^𝑁∶𝑓(𝑡)∈𝐹₁(𝑡,[𝑡,𝑦)a.e.𝑡∈0,𝑏]∶=𝑆_𝐹1,𝑦. (47) It is clear that 𝐹₁(⋅,⋅) is 𝐻_𝑑 continuous and 𝐹₁(⋅,⋅)∈𝒫_𝑤𝑘cpcv(ℝ^𝑁) and is integrably bounded, then by Theorem 17 (see also Theorem 6.5 in [32] or Theorem 1.1 in [34]), we can find a continuous function 𝑔∶𝐾→𝐿¹_𝑤([0,𝑏],ℝ^𝑁) such that 𝑔(𝑥)∈ext𝑆_𝐹1𝑦∀𝑦∈𝐾. (48) From Benamara [49] we know that ext𝑆_𝐹1𝑦=𝑆_{ext𝐹1(⋅,𝑦(⋅))}∀𝑦∈𝐾. (49) Setting 𝑁=𝐿∘𝑔 and letting 𝑦∈𝐾, then 𝑔𝑦∈𝐹₁(⋅)𝑦𝑦𝑔𝑦⋅,𝑦⟹𝑔∈𝑉⟹𝑁=𝐿∈𝐾. (50) Now, we prove that 𝑁 is continuous. Indeed, let 𝑦_𝑛∈𝐾 converge to 𝑦 in 𝐶([0,𝑏],ℝ^𝑁).
Then 𝑔𝑦_𝑛convergeweaklyto𝑔𝑦as𝑛⟶∞. (51) Since 𝑁(𝑦_𝑛)=𝐿(𝑔(𝑦_𝑛))∈𝐾 and 𝑔(𝑦_𝑛)(⋅)∈𝐹(𝑡,𝑦_𝑛(𝑡)), then 𝑔𝑦_𝑛(⋅)∈𝐹⋅,𝐵_𝑀∈𝒫_cpℝ^𝑁. (52) From Lemma 10, 𝑔(𝑦_𝑛) converge weakly to 𝑦 in 𝐿¹([0,𝑏],ℝ^𝑁) as 𝑛→∞. By the definition of 𝑁, we have 𝑁𝑦_𝑛=𝑦₀+𝑦₁1𝑡+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔𝑦_𝑛𝑠)[],𝑁𝑦(𝑑𝑠,𝑡∈0,𝑏=𝑦₀+𝑦₁1𝑡+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔𝑦(𝑠)[].𝑑𝑠,𝑡∈0,𝑏 (53) Since {𝑁(𝑦_𝑛)∶𝑛∈ℕ}⊂𝐾, then there exists subsequence of 𝑁(𝑦_𝑛) converge in 𝐶([0,𝑏],ℝ^𝑁). Then 𝑁𝑦_𝑛(𝑡)⟶𝑁𝑦(𝑡),∀𝑡∈[0,𝑏],as𝑛⟶∞. (54) This proves that 𝑁 is continuous. Hence by Schauder’s fixed point there exists 𝑦∈𝐾 such 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, Theorem 2, page 124] 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]×ℝ^𝑁→𝒫(ℝ^𝑁) be a multifunction satisfying the following hypotheses.(ℋ₃)The function 𝐹∶[0,𝑏]×ℝ^𝑁→𝒫_{𝑐𝑝𝑐𝑣}(ℝ^𝑁) such that, for all 𝑥∈ℝ^𝑁, the map 𝑡⟼𝐹(𝑡,𝑥) (55) is measurable.(ℋ₄)There exists 𝑝∈𝐿¹(𝐽,ℝ⁺) such that 𝐻_𝑑𝐹()(𝑡,[]𝑡,𝑥,𝐹𝑡,𝑦≤𝑝)𝑥−𝑦𝑓𝑜𝑟𝑎.𝑒.𝑡∈0,𝑏𝑎𝑛𝑑𝑒𝑎𝑐ℎ𝑥,𝑦∈ℝ^𝑁,𝐻_𝑑(𝐹(𝑡,0),0)≤𝑝(𝑡)𝑓𝑜𝑟𝑎.𝑒.𝑡∈[0,𝑏]. (56)Then 𝑆_𝑒=𝑆_𝑐.

Proof. By Coviz and Nadlar fixed point theorem, we can easily prove that 𝑆_𝑐≠∅, and since 𝐹 has compact and convex valued, then 𝑆_𝑐 is compact in 𝐶([0,𝑏],ℝ^𝑁). For more information we see [25, 27–29, 51, 52].
Let 𝑦∈𝑆_𝑐; then there exists 𝑓∈𝑆_𝐹,𝑦 such that 𝑦(𝑡)=𝑦₀+𝑦₁1𝑡+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)[].𝑑𝑠,a.e.𝑡∈0,𝑏 (57) Let 𝐾 be a compact and convex set in 𝐶([0,𝑏],ℝ^𝑁) such that 𝑆_𝑐⊂𝐾. Given that 𝑦_∗∈𝐾 and 𝜖>0, we define the following multifunction 𝑈_𝜖∶[0,𝑏]→𝒫(ℝ^𝑁) by 𝑈_𝜖(𝑡)=𝑢∈ℝ^𝑁∶𝑓(𝑡)−𝑢<𝑑𝑓(𝑡),𝐹𝑡,𝑦(𝑡)+𝜖,𝑢∈𝐹𝑡,𝑦_∗(𝑡). (58) The multivalued map 𝑡→𝐹(𝑡,⋅) is measurable and 𝑥→𝐹(⋅,𝑥) is 𝐻_𝑑 continuous. 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],ℝ^𝑁). Then by Lemma 3, there exists a measurable selection 𝑣₁(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e.𝑡∈[0,𝑏] such that 𝑓(𝑡)−𝑣₁(𝑡)<𝑑𝑓(𝑡),𝐹𝑡,𝑦(𝑡)+𝜖; (59) this implies that 𝑈_𝜖(⋅)≠∅. We consider 𝐺_𝜖∶𝐾→𝒫(𝐿¹(𝐽,ℝ^𝑁) defined by 𝐺_𝜖𝑦=𝑓_∗∈ℱ𝑦∶𝑓(𝑡)−𝑓_∗(𝑡𝑓)<𝜖+𝑑_∗(𝑡),𝐹𝑡,𝑦(𝑡). (60) Since the measurable multifunction 𝐹 is integrable bounded, Lemma 9 implies that the Nemyts’kiĭ operator ℱ has decomposable values. Hence 𝑦→𝐺_𝜖(𝑦) is l.s.c. with decomposable values. By Lemma 8, there exists a continuous selection 𝑓_𝜖∶𝐶([0,𝑏],ℝ^𝑁)→𝐿¹(𝐽,ℝ^𝑁) such that 𝑓_𝜖𝑦∈𝐺_𝜖(𝑦)∀𝑦∈𝐶[0,𝑏],ℝ^𝑁. (61) From Theorem 17, there exists function 𝑔_𝜖∶𝐾→𝐿_𝑤([0,𝑏],ℝ^𝑁) such that 𝑔_𝜖𝑦∈ext𝑆_𝐹𝑦=𝑆_{ext𝐹(⋅,𝑦(⋅))}∀𝑦∈𝐾,‖𝑔_𝜖𝑦−𝑓_𝜖𝑦‖_𝑤≤𝜖,∀𝑦∈𝐾. (62) From (ℋ₃) we can prove that there exists 𝑀>0 such that 𝑦_∞≤𝑀∀𝑦∈𝑆_𝑐. (63) Consider the sequence 𝜖_𝑛→0,  as  𝑛→∞, and set 𝑔_𝑛=𝑔_𝜖𝑛, 𝑓_𝑛=𝑓_𝜖𝑛. Set 𝑉=𝑓∈𝐿¹[0,𝑏],ℝ^𝑁∶𝑓(𝑡(𝑡)[,𝜓(𝑡)=()𝑝(𝑡).)≤𝜓a.e.𝑡∈0,𝑏]1+𝑀 (64) Let 𝐿∶𝑉→𝐶([0,𝑏],ℝ^𝑁) be the map such that each 𝑓∈𝑉 assigns the unique solution of the problem ^𝑐𝐷^𝛼𝑦(𝑡)(𝑡)[],𝑦(0)=𝑓,a.e.𝑡∈0,𝑏=𝑦₀,𝑦^′(0)=𝑦₁. (65) As in Theorem 24, we can prove that 𝐿(𝑉) is compact in 𝐶([0,𝑏],ℝ^𝑁) and the operator 𝑁_𝑛=𝐿∘𝑔_𝑛∶𝐾→𝐾 is compact; then by Schauder’s fixed point there exists 𝑦_𝑛∈𝐾 such that 𝑦_𝑛∈𝑆_𝑒 and 𝑦_𝑛(𝑡)=𝑦₀+𝑡𝑦₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔_𝑛𝑦_𝑛𝑠)[](𝑑𝑠,a.e.𝑡∈0,𝑏,𝑛∈ℕ. (66) Hence 𝑦(𝑡)−𝑦_𝑛(𝑡≤1)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔_𝑛𝑦_𝑛(𝑠)(𝑠≤1−𝑓)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔_𝑛𝑦_𝑛(𝑠)−𝑓_𝑛𝑦_𝑛(𝑠+𝑏)𝑑𝑠_𝛼Γ(𝛼)^𝑡₀𝑓_𝑛𝑦_𝑛(𝑠)(𝑠≤1−𝑓)𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔_𝑛𝑦_𝑛(𝑠)−𝑓_𝑛𝑦_𝑛(𝑠+𝑏)𝑑𝑠_𝛼Γ(𝛼)^𝑡₀𝜖_𝑛+𝑑𝑓(𝑠),𝑓_𝑛𝑦_𝑛(𝑠)≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔_𝑛𝑦_𝑛(𝑠)−𝑓_𝑛𝑦_𝑛(𝑠+𝑏)𝑑𝑠_𝛼Γ(𝛼)^𝑡₀𝜖_𝑛+𝐻_𝑑𝐹𝑠,𝑦(𝑠),𝐹𝑠,𝑦_𝑛(𝑠)≤𝑏𝑑𝑠_𝛼+1Γ()𝜖𝛼+1^𝑛+𝑏^𝛼+1Γ(𝛼)𝜖_𝑛+^𝑡₀𝑝(𝑠)𝑦(𝑠)−𝑦_𝑛(𝑠). (67) Let 𝑦(⋅) be a limit point of the sequence 𝑦_𝑛(⋅). Then, it follows that from the above inequality, one has 𝑦(𝑡)−𝑦(𝑡)≤^𝑡₀𝑝(𝑠)𝑦(𝑠)−𝑦(𝑠)𝑑𝑠, (68) which implies 𝑦(⋅)=𝑦(⋅). Consequently, 𝑦∈𝑆_𝑐 is a unique limit point of 𝑦_𝑛(⋅)∈𝑆_𝑒.

Example 26. Let 𝐹∶𝐽×ℝ^𝑁→𝒫_cpcv(ℝ^𝑁) with 𝐹𝑡,𝑦=𝐵𝑓₁𝑡,𝑦,𝑓₂𝑡,𝑦, (69) where 𝑓₁,𝑓₂∶𝐽×ℝ^𝑁→ℝ^𝑁 are Carathéodory functions and bounded.
Then (2) is solvable.

Example 27. If, in addition to the conditions on 𝐹 of Example 26, 𝑓₁ and 𝑓₂ are Lipschitz functions, then 𝑆_𝑒=𝑆_𝑐.

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.