On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval

Abstract

We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion 𝐷^𝛼₀⁺𝑢(𝑡)+𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))∋0,0<𝑡<+∞,𝑢(0)=𝑢^′(0)=0,𝐷^𝛼−1𝑢(+∞)−∑^𝑚−2_𝑖=1𝛽_𝑖𝑢(𝜉_𝑖)=0, where 𝐷^𝛼₀⁺ is the standard Riemann-Liouville fractional derivative, 2<𝛼<3,0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<+∞, satisfies 0<∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1<Γ(𝛼),??and??𝐹∶[0,+∞)×ℝ×ℝ→𝒫(ℝ) is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or nonconvex values.

1. Introduction

In this paper, we will consider the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion 𝐷^𝛼₀⁺𝑢(𝑡)+𝐹𝑡,𝑢(𝑡),𝑢^′(𝑡)𝑢(0)∋0,0<𝑡<+∞,=𝑢^′(0)=0,𝐷^𝛼−1𝑢(+∞)−^𝑚−2_𝑖=1𝛽_𝑖𝑢𝜉_𝑖=0, (1) where 𝐷^𝛼₀⁺ is the standard Riemann-Liouville fractional derivative, 2<𝛼<3, 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<+∞, satisfies 0<∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1<Γ(𝛼), and 𝐹∶[0,+∞)×ℝ×ℝ→𝒫(ℝ) is a set-valued map.

The present paper is motivated by a recent paper of Liang and Zhang [1], where it is considered problem (1) with 𝐹 single valued, and several existence results are provided.

Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering. For details, see [2–4] and the references therein.

The existence of solutions of initial value problems for fractional order differential equations has been studied in the literature [5–17] and the references therein. The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [18]. Also, recently, several qualitative results for fractional differential inclusions were obtained in [19–23] and the references therein.

The aim here is to establish existence results for problem (1) when the right hand side is convex as well as nonconvex valued. In the first result (Theorem 21), we consider the case when the right hand side has convex values and prove an existence result via nonlinear alternative for Kakutani maps. In the second result (Theorem 25), we will use the fixed point theorem for contraction multivalued maps according to Covitz and Nadler. The paper is organized as follows.

In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results. Finally, in Section 4, an example is given to demonstrate the application of one of our main results.

2. Preliminaries

In this section, we present some notations and preliminary lemmas that will be used in the proof of the main result.

Let (𝑋,𝑑) be a metric space with the corresponding norm ‖⋅‖ and let 𝐼=[0,+∞). We denote by ℒ(𝐼) the 𝜎-algebra of all Lebesgue measurable subsets of 𝐼, by ℬ(𝑋) the family of all nonempty subsets of 𝑋, and by 𝒫(𝑋) the family of all Borel subsets of 𝑋. If 𝐴⊂𝐼 then 𝜒_𝐴∶𝐼→{0,1} denotes the characteristic function of 𝐴. For any subset 𝐴⊂𝑋, we denote by 𝐴 the closure of 𝐴.

Recall that the Pompeiu-Hausdorff distance of the closed subsets 𝐴,𝐵⊂𝑋 is defined by the following: 𝑑_𝐻(𝐴,𝐵)=max𝑑^∗(𝐴,𝐵),𝑑^∗(,𝑑𝐵,𝐴)_∗(𝐴,𝐵)=sup{𝑑(𝑎,𝐵),𝑎∈𝐴}, (2) where 𝑑(𝑥,𝐵)=inf_𝑦∈𝐵𝑑(𝑥,𝑦). Define 𝒫(𝑋)={},𝒫𝑌⊂𝑋∶𝑌≠∅_𝑏(𝑋)={(𝑋)},𝒫𝑌∈𝒫∶𝑌bounded_𝑐𝑙(𝑋)={(𝑋)},𝒫𝑌∈𝒫∶𝑌closed_𝑐𝑝(𝑋)=(𝑋),𝒫𝑌∈𝒫∶𝑌compact_𝑐𝑣(𝑋)={𝑌∈𝒫(𝑋)∶𝑌convex}. (3)

Also, we denote by 𝐶(𝐼,𝑋) the Banach space of all continuous functions 𝑥∶[0,+∞)→𝑋 endowed with the norm |𝑥|_𝑐=sup_{𝑡∈[0,+∞)}|𝑥(𝑡)| and by 𝐿¹([0,+∞),𝑋) the Banach space of all (Bochner) integrable functions 𝑥∶[0,+∞)→𝑋 endowed with the norm |𝑥|₁=∫_[0,+∞)|𝑥(𝑡)|𝑑𝑡.

Let (𝑋,𝑑₁) and (𝑌,𝑑₂) be two metric spaces. If 𝑇∶𝑋→𝒫(𝑋) is a set-valued map, then a point 𝑥∈𝑋 is called a fixed point for 𝑇 if 𝑥∈𝑇(𝑥). 𝑇 is said to be bounded on bounded sets if 𝑇(𝐵)∶=∪_𝑥∈𝐵𝑇(𝑥) is a bounded subset of 𝑋 for all bounded sets 𝐵 in 𝑋. 𝑇 is said to be compact if 𝑇(𝐵) is relatively compact for any bounded sets 𝐵 in 𝑋. 𝑇 is said to be totally compact if 𝑇(𝑋) is a compact subset of 𝑋. 𝑇 is said to be upper semicontinuous if for any open set 𝐷⊂𝑋, the set {𝑥∈𝑋∶𝑇(𝑥)⊂𝐷} is open in 𝑋. 𝑇 is called completely continuous if it is upper semicontinuous and, for every bounded subset 𝐴⊂𝑋, 𝑇(𝐴) is relatively compact. It is well known that a compact set-valued map 𝑇 with nonempty compact values is upper semicontinuous if and only if 𝑇 has a closed graph.

We define the graph of 𝑇 to be the set Gr(𝑇)={(𝑥,𝑦)∈𝑋×𝑌,𝑦=𝑇(𝑥)} and recall a useful result regarding connection between closed graphs and upper semicontinuity.

Lemma 1 (see [24, Proposition 1.2]). If 𝑇∶𝑋→𝒫_𝑐𝑙(𝑌) is upper semicontinuous, then Gr(𝑇) is a closed subset of 𝑋×𝑌, that is, for every sequence {𝑥_𝑛}_𝑛∈ℕ⊂𝑋 and {𝑦_𝑛}_𝑛∈ℕ⊂𝑌, if when 𝑛→∞, 𝑥_𝑛→𝑥_∗, 𝑦_𝑛→𝑦_∗, and 𝑦_𝑛∈𝑇(𝑥_𝑛), then 𝑦_∗∈𝑇(𝑥_∗). Conversely, if 𝑇 is completely continuous and has a closed graph, then it is upper semicontinuous.

For the convenience of the reader, we present here the following nonlinear alternative of the Leray-Schauder type and its consequences.

Theorem 2 (nonlinear alternative for Kakutani maps [25]). Let 𝑋 be a Banach space, 𝐶 a closed convex subset of 𝑋, 𝑈 an open subset of 𝐶, and 0∈𝑈. Suppose that 𝑇∶𝑈→𝒫_{𝑐𝑙,𝑐𝑣}(𝐶) is an upper semicontinuous compact map; here 𝒫_{𝑐𝑙,𝑐𝑣}(𝐶) denotes the family of nonempty, compact convex subsets of 𝐶. Then, either(i)𝑇has a fixed point in 𝑈 or(ii)there is a 𝑢∈𝜕𝑈 and 𝜆∈(0,1) with 𝑢∈𝜆𝑇(𝑢).

Definition 3. The multifunction 𝑇∶𝑋→𝒫(𝑋) is said to be lower semicontinuous if for any closed subset 𝐶⊂𝑋, the subset {𝑠∈𝑋∶𝑇(𝑠)⊂𝐶} is closed.

If 𝐹∶[0,+∞)×ℝ×ℝ→𝒫(ℝ) is a set-valued map with compact values and 𝑥∈𝐶([0,+∞),ℝ), we define 𝑆_𝐹(𝑥)∶=𝑓∈𝐿¹))(𝑡)(𝑡)([0,+∞,ℝ∶𝑓∈𝐹𝑡,𝑥,𝑥^′(𝑡)a.e.[0,+∞). (4) Then, 𝐹 is of a lower semicontinuous type if 𝑆_𝐹(⋅) is a lower semicontinuous with closed and decomposable values.

Theorem 4 (see [26]). Let 𝑆 be a separable metric space and 𝐺∶𝑆→𝒫(𝐿¹([0,+∞),ℝ)) be a lower semicontinuous set-valued map with closed decomposable values. Then 𝐺 has a continuous selection (i.e., there exists a continuous mapping 𝑔∶𝑆→𝐿¹([0,+∞),ℝ) such that 𝑔(𝑠)∈𝐺(𝑠) for all 𝑠∈𝑆).

Definition 5. Consider the following.(i)A set-valued map 𝐺∶[0,+∞)→𝒫(ℝ) with nonempty compact convex values is said to be measurable if for any 𝑥∈ℝ the function 𝑡→𝑑(𝑥,𝐺(𝑡)) is measurable.(ii)A set-valued map 𝐹∶[0,+∞)×ℝ×ℝ→𝒫(ℝ) is said to be Carathéodory if 𝑡→𝐹(𝑡,𝑥,𝑦) is measurable for all 𝑥,𝑦∈ℝ and (𝑥,𝑦)→𝐹(𝑡,𝑥,𝑦) is upper semicontinuous for almost all 𝑡∈[0,+∞).(iii)𝐹 is said to be 𝐿¹-Carathéodory if for any 𝑙>0 there exists ℎ_𝑙∈𝐿¹([0,+∞),ℝ) such that sup{|𝑣|∶𝑣∈𝐹(𝑡,𝑥,𝑦)}≤ℎ_𝑙(𝑡)a.e.[0,+∞);∀𝑥,𝑦∈ℝ.

Finally, the following results are easily deduced from the theoretical limit set properties.

Lemma 6 (see [27, Lemma 1.1.9]). Let {𝐾_𝑛}_𝑛∈ℕ⊂𝐾⊂𝑋 be a sequence of subsets where 𝐾 is a compact subset of a separable Banach space 𝑋. Then, colimsup_𝑛→∞𝐾_𝑛=_𝑁>0co_𝑛≥𝑁𝐾_𝑛, (5) where co(𝐴) refers to the closure of the convex hull of 𝐴.

Lemma 7 (see [27, Lemma 1.4.13]). Let 𝑋 and 𝑌 be two metric spaces. If 𝐺∶𝑋→𝒫_𝑐𝑝(𝑌) is an upper semicontinuous, then, for each 𝑥₀∈𝑋, limsup_𝑥→𝑥0𝐺(𝑥)=𝐺𝑥₀. (6)

Definition 8. Let 𝑋 be a Banach space. A sequence {𝑥_𝑛}_𝑛∈ℕ⊂𝐿¹([𝑎,𝑏],𝑋) is said to be semicompact if(a)it is integrably bounded; that is, there exists 𝑞∈𝐿¹([𝑎,𝑏],ℝ⁺) such that 𝑥_𝑛(𝑡)_𝐸≤𝑞(𝑡),fora.e.𝑡∈[𝑎,𝑏]andevery𝑛∈ℕ, (7)(b)the image sequence {𝑥_𝑛(𝑡)}_𝑛∈ℕ is relatively compact in 𝐸 for a.e. 𝑡∈[𝑎,𝑏].

The following important result follows from the Dunford-Pettis theorem (see [28, Proposition 4.2.1]).

Lemma 9. Every semicompact sequence 𝐿¹([𝑎,𝑏],𝑋) is weakly compact in 𝐿¹([𝑎,𝑏],𝑋).

When the nonlinearity takes convex values, Mazur’s Lemma, 1933, may be useful.

Lemma 10 (see [29, Theorem 21.4]). Let 𝐸 be a normed space and {𝑥_𝑘}_𝑘∈ℕ⊂𝐸 a sequence weakly converging to a limit 𝑥∈𝐸. Then, there exists a sequence of convex combinations 𝑦_𝑚=∑^𝑚_𝑘=1𝛼_𝑚𝑘𝑥_𝑘 with 𝛼_𝑚𝑘>0 for 𝑘=1,2,…,𝑚 and ∑^𝑚_𝑘=1𝛼_𝑚𝑘=1 which converges strongly to 𝑥.

Lemma 11 (see [30]). Let 𝑋 be defined as before and 𝑀⊂𝑋. Then 𝑀 is relatively compact in 𝑋 if the following conditions hold:(a)𝑀 is uniformly bounded in 𝑋;(b)the functions from 𝑀 are equicontinuous on any compact interval of [0,+∞);(c)the functions from 𝑀 are equiconvergent; that is, for any given 𝜖>0, there exists a 𝑇=𝑇(𝜖)>0 such that |𝑓(𝑡)−𝑓(+∞)|<𝜖, for any 𝑡>𝑇, 𝑓∈𝑀.

Definition 12 (see [6]). The Riemann-Liouville fractional integral operator of order 𝛼>0, of function 𝑓∈𝐿¹(ℝ⁺), is defined as 𝐼^𝛼₀⁺𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠, (8) where Γ(⋅) is the Euler gamma function.

Definition 13 (see [31]). The Riemann-Liouville fractional derivative of order 𝛼>0, 𝑛−1<𝛼<𝑛, 𝑛∈ℕ is defined as 𝐷^𝛼₀⁺𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)𝑑𝑠, (9) where the function 𝑓(𝑡) has absolutely continuous derivatives up to order (𝑛−1).

Lemma 14 (see [31]). The equality 𝐷^𝛾₀₊𝐼^𝛾₀₊𝑓(𝑡)=𝑓(𝑡), 𝛾>0 holds for 𝑓∈𝐿¹(0,1).

Lemma 15 (see [31]). Let 𝛼>0 and 𝑢∈𝐶(0,1)∩𝐿¹(0,1). Then, the differential equation 𝐷^𝛼₀⁺𝑢(𝑡)=0 (10) has a unique solution 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, 𝑐_𝑖∈ℝ, 𝑖=1,…,𝑛, where 𝑛−1<𝛼<𝑛.

Lemma 16 (see [31]). Let 𝛼>0. Then, the following equality holds for 𝑢∈𝐿¹(0,1), 𝐷^𝛼₀⁺𝑢∈𝐿¹(0,1): 𝐼^𝛼₀⁺𝐷^𝛼₀⁺𝑢(𝑡)=𝑢(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛, (11)𝑐_𝑖∈ℝ, 𝑖=1,…,𝑛, where 𝑛−1<𝛼≤𝑛.

By 𝐴𝐶¹([0,+∞),ℝ) we denote the space of continuous real-valued functions whose first derivative exists and it is absolutely continuous on [0,+∞). In this paper, we will use the following space 𝐸 to the study (1) which is denoted by 𝐸=𝑢∈𝐴𝐶¹([0,+∞),ℝ)∶sup_{0≤𝑡<+∞}|𝑢(𝑡)|1+𝑡^𝛼−1,sup_{0≤𝑡<+∞}𝑢^′(𝑡)1+𝑡^𝛼−1<+∞. (12)

From [32], we know that 𝐸 is a Banach space equipped with the norm ‖𝑢‖=maxsup_{0≤𝑡<+∞}|𝑢(𝑡)|1+𝑡^𝛼−1,sup_{0≤𝑡<+∞}𝑢^′(𝑡)1+𝑡^𝛼−1. (13)

In what follows, 𝐼=[0,+∞), 𝛼∈(2,3), and Δ=∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1. Next, we need the following technical result proved in [1].

Lemma 17 (see [1]). For any ℎ∈𝐿¹([0,+∞),ℝ), the problem 𝐷^𝛼₀⁺𝑢(𝑡)(𝑡)𝑢(0)+ℎ=0,0<𝑡<∞,2<𝛼<3,=𝑢^′(0)=0,𝐷^𝛼−1𝑢(+∞)=^𝑚−2_𝑖=1𝛽_𝑖𝑢𝜉_𝑖, (14) has a unique solution 𝑢(𝑡) that 𝑢(𝑡)=₀^+∞𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (15) where 𝐺(𝑡,𝑠)=𝐺₁(𝑡,𝑠)+𝐺₂(),𝐺𝑡,𝑠₁()=1𝑡,𝑠Γ(𝛼)𝑡^𝛼−1−(𝑡−𝑠)^𝛼−1𝑡,0≤𝑠≤𝑡<+∞,_𝛼−1𝐺,0≤𝑡≤𝑠<+∞,₂()=∑𝑡,𝑠^{𝑚−2𝑖=1}𝛽_𝑖𝑡^𝛼−1Γ(𝛼)𝐺−Δ¹𝜉_𝑖,𝑠. (16)(17)(18)

Note that 𝐺(𝑡,𝑠)>0,∀𝑡,𝑠∈[0,+∞), (e.g., Lemma??3.2 in [1]) and from the definition of 𝐺₁(𝑡,𝑠), we have the following (e.g., Remark??3.1 in [1]): 𝐺₁(𝑡,𝑠)1+𝑡^𝛼−1≤1Γ(𝛼),𝐺(𝑡,𝑠)1+𝑡^𝛼−1≤𝐿₁for(𝑡,𝑠)∈[0,+∞)×[0,+∞), (19) where 𝐿₁=1Γ(𝛼)∑1+^{𝑚−2𝑖=1}𝛽_𝑖𝜉^𝛼−1_𝑚−2(Γ(𝛼)).−Δ (20) Also, one can get 𝜕𝐺(𝑡,𝑠)/𝜕𝑡1+𝑡^𝛼−1≤𝐿₂for(𝑡,𝑠)∈[0,+∞)×[0,+∞), (21) where 𝐿₂=2(𝛼−1)Γ(𝛼)∑1+^{𝑚−2𝑖=1}𝛽_𝑖𝜉^𝛼−1_𝑚−2(Γ(𝛼)).−Δ (22)

Lemma 18. The function 𝐺(𝑡,𝑠) defined by (16) satisfies lim_𝑡→+∞𝐺(𝑡,𝑠)1+𝑡^𝛼−1=∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−1Γ(𝛼Γ(𝛼)).)(−Δ (23)

By calculation, it is easy to prove that Lemma 18 holds. So, we omit its proof here.

3. Main Results

Now we are able to present the existence results for problem (1).

3.1. The Upper Semicontinuous Case

To obtain the complete continuity of existence solutions operator, the following lemma is still needed.

Lemma 19 (see [32]). Let 𝑉={𝑢∈𝐸∣‖𝑢‖<𝑙}(𝑙>0),𝑉₁={𝑢(𝑡)/(1+𝑡^𝛼−1)∶𝑢∈𝑉}. If?? 𝑉₁ is equicontinuous on any compact interval of [0,+∞) and equiconvergent at infinity, then 𝑉 is relatively compact on 𝐸.

Definition 20. 𝑉₁ is called equiconvergent at infinity if and only if for all 𝜖>0, there exists 𝑣(𝜖)>0 such that for all 𝑢∈𝑉₁,𝑡₁,𝑡₂≥𝑣, it holds 𝑢𝑡₁1+𝑡₁^𝛼−1−𝑢𝑡₂1+𝑡₂^𝛼−1<𝜖. (24)

Theorem 21. The Carathéodory multivalued map 𝐹∶𝐼×ℝ×ℝ→𝒫(ℝ) has nonempty, compact, convex values and satisfies the following.(H1)?? There exists a continuous nondecreasing function 𝜓∶[0,+∞)→(0,+∞) and 𝜑∈𝐿¹([0,+∞), ℝ⁺) such that‖𝐹(𝑡,𝑥,𝑦)‖_𝒫∶=sup{|𝑣(𝑡)/(1+𝑡^𝛼−1)|∶𝑣∈𝐹(𝑡,𝑥,𝑦)}≤𝜑(𝑡)𝜓(‖𝑥‖),fora.e.𝑡∈𝐼??and??each??𝑥,𝑦∈ℝ.(H2) There exists a constant 𝑀>0 such that 𝑀max𝐿₁,𝐿₂𝜓(𝑀)∫₀^+∞𝜑(𝑠)𝑑𝑠>1. (25)
Then, problem (1) has at least one solution.

Proof. Let 𝑋=𝐸 and consider 𝑀>0 as in (25). It is obvious that the existence of solutions to problem (1) is reduced to the existence of the solutions of the integral inclusion 𝑢(𝑡)∈₀^+∞𝐺(𝑡,𝑠)𝐹𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠,𝑡∈𝐼, (26) where 𝐺(𝑡,𝑠) is defined by (16) and (17). Consider the set-valued map, 𝑇∶𝐸→𝒫(𝑋) is defined by 𝑇(𝑢)∶=𝑣∈𝑋;𝑣(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠,𝑓∈𝑆_𝐹(𝑢). (27)
We show that 𝑇 satisfies the hypotheses of Theorem 2.
Claim 1. We show that 𝑇(𝑢)⊂𝑋 is convex for any 𝑢∈𝑋. If 𝑣₁,𝑣₂∈𝑇(𝑢), then, there exist 𝑓₁,𝑓₂∈𝑆_𝐹(𝑢) such that for any 𝑡∈𝐼 one has 𝑣_𝑖(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓_𝑖(𝑠)𝑑𝑠,𝑖=1,2. (28) Let 0≤𝜆≤1. Then, for any 𝑡∈𝐼, we have 𝜆𝑣₁+(1−𝜆)𝑣₂(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝜆𝑓₁(𝑠)+(1−𝜆)𝑓₂(𝑠)𝑑𝑠. (29) The values of 𝐹 are convex; thus, 𝑆_𝐹(𝑢) is a convex set and hence 𝜆𝑣₁+(1−𝜆)𝑣₂∈𝑇(𝑢).
Claim 2. We show that 𝑇 is bounded on bounded sets of 𝑋. Let 𝐵 be any bounded subset of 𝑋. Then, there exists 𝑚>0 such that ‖𝑢‖≤𝑚 for all 𝑢∈𝐵. If 𝑣∈𝑇(𝑢), then there exists 𝑓∈𝑆_𝐹(𝑢) such that 𝑣(𝑡)=∫₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. One may write the following for any 𝑡∈𝐼: 𝑣(𝑡)1+𝑡^𝛼−1≤₀^+∞𝐺(𝑡,𝑠)1+𝑡^𝛼−1𝑓(𝑠)𝑑𝑠≤𝐿₁₀^+∞𝜑(𝑠)𝜓(‖𝑢‖)𝑑𝑠≤𝐿₁𝜑₁𝜓(𝑚). (30) On the other hand, 𝑣^′(𝑡)1+𝑡^𝛼−1≤₀^+∞𝜕𝐺(𝑡,𝑠)/𝜕𝑡1+𝑡^𝛼−1𝑓(𝑠)𝑑𝑠≤𝐿₂₀^+∞𝜑(𝑠)𝜓(‖𝑢‖)𝑑𝑠≤𝐿₂𝜑₁𝜓(𝑚), (31) and therefore ‖𝑣‖=max_𝑡∈𝐼𝑣(𝑡)1+𝑡^𝛼−1,𝑣^′(𝑡)1+𝑡^𝛼−1≤max𝐿₁,𝐿₂𝜑₁𝜓(𝑚), (32) for all 𝑣∈𝑇(𝑢); that is, 𝑇(𝐵) is bounded.
Claim 3. We show that 𝑇 maps bounded the sets into equi-continuous sets. Let 𝐵 be any bounded subset of 𝑋 as before and 𝑣∈𝑇(𝑢) for some 𝑢∈𝐵. Then, there exists 𝑓∈𝑆_𝐹(𝑢) such that 𝑣(𝑡)=∫₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. So, for any 𝑇₀∈(0,+∞) and 𝑡₁,𝑡₂∈[0,𝑇₀], without loss of generality, we may assume that 𝑡₂>𝑡₁ and one can get the following: 𝑣𝑡₁1+𝑡₁^𝛼−1−𝑣𝑡₂1+𝑡₂^𝛼−1≤₀^+∞𝐺₁𝑡₁,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₂^𝛼−1𝑓(𝑠)+∑𝑑𝑠_{𝑚−2𝑖=1}𝛽_𝑖Γ(𝛼)𝑡−Δ1+𝑡^1𝛼−1₁^𝛼−1−𝑡₂^𝛼−11+𝑡₂^𝛼−1×₀^+∞𝐺₁𝜉_𝑖𝑓(𝑠)≤,𝑠𝑑𝑠_0+∞𝐺₁𝑡₁,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₁^𝛼−1𝜑(𝑠)𝜓𝑢+(‖‖)𝑑𝑠_0+∞𝐺₁𝑡₂,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₂^𝛼−1𝜑(𝑠)𝜓𝑢+∑(‖‖)𝑑𝑠_{𝑚−2𝑖=1}𝛽_𝑖𝜉^𝛼−1_𝑚−2Γ(Γ(𝛼)𝑡−Δ1+𝑡^1𝛼−1₁^𝛼−1−𝑡₂^𝛼−11+𝑡₂^𝛼−1×₀^+∞𝜑(𝑠)𝜓(‖𝑢‖)𝑑𝑠. (33)
On the other hand, we get ₀^+∞𝐺₁𝑡₁,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₁^𝛼−1𝜑(𝑠)𝜓𝑢≤(‖‖)𝑑𝑠_𝑡10+^𝑡2_𝑡1+_𝑡^+∞₂𝐺₁𝑡₁,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₁^𝛼−1(𝑠)𝜓𝑢(𝑚)×𝜑(‖‖)𝑑𝑠≤𝜓^𝑡1₀𝑡₂^𝛼−1−𝑡₁^𝛼−1+𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−11+𝑡₁^𝛼−1(𝑠)(𝑚)×𝜑𝑑𝑠+𝜓^𝑡2_𝑡1𝑡₂^𝛼−1−𝑡₁^𝛼−1+𝑡₂−𝑠^𝛼−11+𝑡₁^𝛼−1𝜑(𝑠)(𝑚)𝑑𝑠+𝜓_𝑡^+∞₂𝑡₂^𝛼−1−𝑡₁^𝛼−11+𝑡₁^𝛼−1𝜑(𝑠)𝑑𝑠→0uniformlyas𝑡₁→𝑡₂. (34) Similar to (34), we have ₀^+∞𝐺₁𝑡₂,𝑠1+𝑡₁^𝛼−1−𝐺₁𝑡₂,𝑠1+𝑡₂^𝛼−1𝜑(𝑠)𝜓(‖𝑢‖)𝑑𝑠→0uniformlyas𝑡₁→𝑡₂. (35) From (34) and (35), we have 𝑣𝑡₁1+𝑡₁^𝛼−1−𝑣𝑡₂1+𝑡₂^𝛼−1→0uniformlyas𝑡₁→𝑡₂. (36) Similar to (36), one can get 𝑣^′𝑡₁1+𝑡₁^𝛼−1−𝑣^′𝑡₂1+𝑡₂^𝛼−1→0uniformlyas𝑡₁→𝑡₂. (37) Therefore, 𝑇(𝐵) is an equicontinuous set in 𝑋.
Claim 4. We show that 𝑇 is equiconvergent at ∞. Let 𝑣∈𝑇(𝑢) for some 𝑢∈𝐵. Then, there exists 𝑓∈𝑆_𝐹(𝑢) such that 𝑣(𝑡)=∫₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. So, we have the following: ₀^+∞𝑓(𝑠)𝑑𝑠≤𝜓(𝑚)₀^+∞𝜑(𝑠)𝑑𝑠<+∞,lim_𝑡→+∞𝑣(𝑡)1+𝑡^𝛼−1=lim_𝑡→+∞11+𝑡^𝛼−1₀^+∞𝐺()𝑓(𝑠)=∑𝑡,𝑠𝑑𝑠_{𝑚−2𝑖=1}𝛽_𝑖𝜉_𝑖^𝛼−1Γ(𝛼)(Γ(𝛼)−Δ)₀^+∞𝑓(𝑠)−∑𝑑𝑠_{𝑚−2𝑖=1}𝛽_𝑖Γ(𝛼)(Γ(𝛼)−Δ)×^𝜉𝑖₀𝜉_𝑖−𝑠^𝛼−1𝑓(𝑠)𝑑𝑠<∞, (38) and, similarly, one has lim_𝑡→+∞𝑣^′(𝑡)1+𝑡^𝛼−1<∞. (39) Therefore, 𝑇(𝐵) is equiconvergent at infinity.
Therefore, with Lemma 11, Lemma 19 and Claims 2–4, we conclude that 𝑇 is completely continuous.
Claim 5. 𝑇 is upper semicontinuous. To this end, it is sufficient to show that 𝑇 has a closed graph. Let 𝑣_𝑛∈𝑇(𝑢_𝑛) such that 𝑣_𝑛→𝑣 and 𝑢_𝑛→𝑢, as 𝑛→+∞. Then, there exists 𝑚>0 such that ‖𝑢_𝑛‖≤𝑚. We will prove that 𝑣∈𝑇(𝑢) means that there exists 𝑓_𝑛∈𝑆_𝐹(𝑢_𝑛) such that, for a.e. 𝑡∈𝐼, we have 𝑣_𝑛(𝑡)=∫₀^+∞𝐺(𝑡,𝑠)𝑓_𝑛(𝑠)𝑑𝑠. Then, we need to show that 𝑣∈𝑇(𝑢).
Condition (H1) implies that 𝑓_𝑛(𝑡)∈𝜑(𝑡)𝜓(𝑚)𝐵₁(0). Then, {𝑓_𝑛}_𝑛∈ℕ is integrably bounded in 𝐿¹(𝐼,ℝ). Since 𝐹 has compact values, we deduce that {𝑓_𝑛}_𝑛 is semicompact. By Lemma 9, there exists a subsequence, still denoted as {𝑓_𝑛}_𝑛∈ℕ, which converges weakly to some limit 𝑓∈𝐿¹(𝐼,ℝ). Moreover, the mapping Γ∶𝐿¹(𝐼,ℝ)→𝑋=𝐸 defined by Γ𝑔(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑔(𝑠)𝑑𝑠 (40) is a continuous linear operator. Then, it remains continuous if these spaces are endowed with their weak topologies [29, 33]. Moreover, for a.e. 𝑡∈𝐼, 𝑢_𝑛(𝑡) converges to 𝑢(𝑡). Then, we have 𝑣(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. (41) It remains to prove that 𝑓∈𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡)), a.e. 𝑡∈𝐼. Mazur’s Lemma (see Lemma 10) yields the existence of 𝛼^𝑛_𝑖≥0, 𝑖=𝑛,…,𝑘(𝑛) such that ∑^𝑘(𝑛)_𝑖=1𝛼^𝑛_𝑖=1 and the sequence of convex combinations 𝑔_𝑛(⋅)=∑^𝑘(𝑛)_𝑖=1𝛼^𝑛_𝑖𝑓_𝑖(⋅) converges strongly to 𝑓 in 𝐿¹. Using Lemma 6, we obtain that 𝑣(𝑡)∈_𝑛≥1𝑔_𝑛(𝑡⊂),a.e.𝑡∈𝐼_𝑛≥1co𝑓_𝑘(𝑡)⊂,𝑘≥𝑛_𝑛≥1co_𝑛≥1𝐹𝑡,𝑢_𝑘(𝑡),𝑢^′_𝑘(𝑡)=colimsup_𝑘→+∞𝐹𝑡,𝑢_𝑘(𝑡),𝑢^′_𝑘(𝑡). (42) However, the fact that the multivalued 𝑥→𝐹(⋅,𝑥,𝑥^′) is upper semicontinuous and has compact values, together with Lemma 7, implies that limsup_𝑛→+∞𝐹𝑡,𝑢_𝑛(𝑡),𝑢^′_𝑛(𝑡)=𝐹𝑡,𝑢(𝑡),𝑢^′(𝑡),a.e.𝑡∈𝐼. (43) This along with (42) yields that 𝑓(𝑡)∈co𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡)). Finally, 𝐹(⋅,⋅,⋅) has closed, convex values; hence, 𝑓(𝑡)∈𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡)), a.e. 𝑡∈𝐼. Thus, 𝑣∈𝑇(𝑢), proving that 𝑇 has a closed graph. Finally, with Lemma 1 and the compactness of 𝑇, we conclude that 𝑇 is upper semicontinuous.
Claim 6. A priori bounds on solutions. Let 𝑢 be a solution of (1). Then, there exists 𝑓∈𝐿¹([0,+∞),ℝ) with 𝑓∈𝑆_𝐹(𝑢) such that 𝑢(𝑡)=∫₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠. In view of (H1), and using the computations in Claim 2 above, for each 𝑡∈[0,+∞), we obtain 𝑢(𝑡)1+𝑡^𝛼−1,𝑢^′(𝑡)1+𝑡^𝛼−1≤max𝐿₁,𝐿₂₀^+∞𝑓(𝑠)𝐿𝑑𝑠≤max₁,𝐿₂𝜓𝑢×(‖‖)_0+∞𝜑(𝑠)𝑑𝑠. (44) Consequently, ‖𝑢‖max𝐿₁,𝐿₂𝜓(‖𝑢‖)∫₀^+∞𝜑(𝑠)𝑑𝑠≤1. (45)
In view of (H2), there exists 𝑀 such that ‖𝑢‖≠𝑀. Let us consider the following: 𝑈∶=𝑢∈𝐴𝐶¹([0,+∞),ℝ)∶‖𝑢‖<𝑀. (46)
Note that the operator 𝑇∶𝑈→𝒫(𝐴𝐶¹([0,+∞)) is upper semicontinuous and completely continuous. From the choice of 𝑈, there is no 𝑢∈𝜕𝑈 such that 𝑢=𝜆𝑇(𝑢) for some 𝜆∈(0,1). Consequently, by the nonlinear alternative of the Leray-Schauder type (Theorem 2), we deduce that 𝑇 has a fixed point 𝑢∈𝑈 which is a solution of the problem (1). This completes the proof.

3.2. The Lipschitz Case

Now we prove the existence of solutions for the problem (1) with a nonconvex-valued right hand side by applying a fixed point theorem for multivalued maps according to Covitz and Nadler [34].

Definition 22. A multivalued operator 𝑁∶𝑋→𝒫_𝑐𝑙(𝑋) is called the following:(a)𝛾-Lipschitz if and only if there exists 𝛾>0 such that 𝑑_𝐻(𝑁(𝑥),𝑁(𝑦))≤𝛾𝑑(𝑥,𝑦) for each 𝑥,𝑦∈𝑋;(b)a contraction if and only if it is 𝛾-Lipschitz with 𝛾<1.

Lemma 23 (Covitz-Nadler, [34]). Let (𝑋,𝑑) be a complete metric space. If 𝑁∶𝑋→𝒫_𝑐𝑙(𝑋) is a contraction, then 𝐹𝑖𝑥𝑁≠∅.

Definition 24. A measurable multivalued function 𝐹∶[0,+∞)→𝒫(𝑋) is said to be integrably bounded if there exists a function 𝑓∈𝐿¹([0,+∞),𝑋) such that for all 𝑣∈𝐹(𝑡), ‖𝑣‖≤𝑓(𝑡) for a.e. 𝑡∈[0,+∞).

Theorem 25. Assume that the following condition holds:(H4) ??𝐹∶𝐼×ℝ×ℝ→𝒫_𝑐𝑝(ℝ) such that 𝐹(⋅,𝑥,𝑦)∶[0,+∞)→𝒫_𝑐𝑝(ℝ) is measurable for each 𝑥,𝑦∈ℝ;(H5) There exist 𝑙₁,𝑙₂∶[0,+∞)→[0,+∞) which are not identical zero on any closed subinterval of [0,+∞), and ₀^+∞1+𝑠^𝛼−1𝑙_𝑖(𝑠)𝑑𝑠<+∞,𝑖=1,2, (47)such that for almost all 𝑡∈[0,+∞), 𝑑_𝐻𝐹𝑡,𝑥₁,𝑦₁,𝐹𝑡,𝑥₂,𝑦₂≤𝑙₁(𝑡)𝑥₁−𝑥₂+𝑙₂(𝑡)𝑦₁−𝑦₂ (48) for all 𝑥₁,𝑥₂,𝑦₁, and 𝑦₂∈ℝ with 𝑑(0,𝐹(𝑡,0,0))≤𝑙₁(𝑡)+𝑙₂(𝑡) for almost all 𝑡∈[0,+∞).
Then, the boundary value problem (1) has at least one solution on 𝐼=[0,+∞) if max𝐿₁,𝐿₂₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠)+𝑙₂(𝑠)𝑑𝑠<1. (49)

Proof. We transform problem (1) into a fixed point problem. Consider the set-valued map 𝑇∶𝐴𝐶¹[0,+∞)→𝒫(𝐴𝐶¹[0,+∞)) defined at the beginning of the proof of Theorem 21. It is clear that the fixed point of 𝑇 are solutions of the problem (1).
Note that since the set-valued map 𝐹(⋅,𝑢(⋅)) is measurable with the measurable selection theorem (e.g., Theorem III.6 in [35]) it admits a measurable selection 𝑓∶𝐼→ℝ. Moreover, since 𝐹 is integrably bounded, 𝑓∈𝐿¹([0,+∞),ℝ). Therefore, 𝑆_𝐹(𝑢)≠∅.
We will prove that 𝑇 fulfills the assumptions of Covitz-Nadler contraction principle (Lemma 23).
First, we note that since 𝑆_𝐹(𝑢)≠∅, 𝑇(𝑢)≠∅ for any 𝑢∈𝐴𝐶¹([0,+∞),ℝ).
Second, we prove that 𝑇(𝑢) is closed for any 𝑢∈𝐴𝐶¹([0,+∞),ℝ). Let {𝑢_𝑛}_𝑛≥0∈𝑇(𝑢) such that 𝑢_𝑛→𝑢₀ in 𝐴𝐶¹([0,+∞),ℝ). Then 𝑢₀∈𝐴𝐶¹([0,+∞),ℝ) and there exists 𝑓_𝑛∈𝑆_𝐹(𝑢) such that 𝑢_𝑛(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓_𝑛(𝑠)𝑑𝑠. (50) Since 𝐹 has compact values, we may pass onto a subsequence (if necessary) to obtain that 𝑓_𝑛 converges to 𝑓∈𝐿¹(([0,+∞),ℝ)) in 𝐿¹(([0,+∞),ℝ)). In particular, 𝑓∈𝑆_𝐹(𝑢), and for any 𝑡∈[0,+∞) we have 𝑢_𝑛(𝑡)→𝑢₀(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓(𝑠)𝑑𝑠, (51) that is, 𝑢₀∈𝑇(𝑢) and 𝑇(𝑢) is closed.
Next we show that 𝑇 is a contraction on 𝐴𝐶¹([0,+∞),ℝ). Let 𝑢₁,𝑢₂∈𝐴𝐶¹([0,+∞),ℝ) and 𝑣₁∈𝑇(𝑢₁). Then there exist 𝑓₁∈𝑆_𝐹(𝑢₁) such that 𝑣₁(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓₁(𝑠)𝑑𝑠,𝑡∈[0,+∞). (52) Consider the set-valued map 𝐻(𝑡)∶=𝐹𝑡,𝑢₂(𝑡),𝑢^′₂(𝑡)∩𝑢∈ℝ;𝑓₁(𝑡)−𝑢≤𝑙₁(𝑡)𝑥₁−𝑥₂+𝑙₂(𝑡)𝑥^′₁−𝑥^′₂,𝑡∈[0,+∞). (53) By (H5), we have 𝑑_𝐻𝐹𝑡,𝑥₁,𝑥^′₁,𝐹𝑡,𝑥₂,𝑥^′₂≤𝑙₁(𝑡)𝑥₁−𝑥₂+𝑙₂(𝑡)𝑥^′₁−𝑥^′₂, (54) hence 𝐻 has nonempty closed values. Moreover, since 𝐻 is measurable (e.g., Proposition III.4 in [35]), there exists 𝑓₂ which is a measurable selection of 𝐻. It follows that 𝑓₂∈𝑆_𝐹(𝑢₂) and for any 𝑡∈[0,+∞), 𝑓₁(𝑡)−𝑓₂(𝑡)≤𝑙₁(𝑡)𝑥₁−𝑥₂+𝑙₂(𝑡)𝑥^′₁−𝑥^′₂. (55) Define 𝑣₂(𝑡)=₀^+∞𝐺(𝑡,𝑠)𝑓₂(𝑠)𝑑𝑠,𝑡∈[0,+∞), (56) and one can get 𝑣₁(𝑡)1+𝑡^𝛼−1−𝑣₂(𝑡)1+𝑡^𝛼−1≤₀^+∞𝐺(𝑡,𝑠)1+𝑡^𝛼−1𝑓₁(𝑠)−𝑓₁(𝑠)𝑑𝑠≤𝐿₁₀^+∞𝑙₁(𝑠)𝑥₁(𝑠)−𝑥₂(𝑠)+𝑙₂(𝑠)𝑥^′₁(𝑠)−𝑥^′₂(𝑠)𝑑𝑠≤𝐿₁₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠𝑥)¹(𝑠)−𝑥₂(𝑠)1+𝑠^𝛼−1+𝑙₂(𝑠𝑥)^′1(𝑠)−𝑥^′₂(𝑠)1+𝑠^𝛼−1𝐿𝑑𝑠≤max₁,𝐿₂𝑥₁−𝑥₂₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠)+𝑙₂(𝑠)𝑑𝑠. (57) Similarly, we have 𝑣^′₁(𝑡)1+𝑡^𝛼−1−𝑣^′₂(𝑡)1+𝑡^𝛼−1≤max𝐿₁,𝐿₂𝑥₁−𝑥₂×₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠)+𝑙₂(𝑠)𝑑𝑠. (58) Therefore, 𝑣₁−𝑣₂≤max𝐿₁,𝐿₂𝑥₁−𝑥₂×₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠)+𝑙₂(𝑠)𝑑𝑠. (59) From an analogous reasoning by interchanging the roles of 𝑢₁ and 𝑢₂, it follows that 𝑑_𝐻𝑇𝑢₁,𝑇𝑢₂≤max𝐿₁,𝐿₂𝑥₁−𝑥₂×₀^+∞1+𝑠^𝛼−1𝑙₁(𝑠)+𝑙₂(𝑠)𝑑𝑠. (60) Since 𝑇 is a contraction, it follows by the Lemma 23 that 𝑇 admits a fixed point which is a solution to problem (1).

4. Application

Consider the fractional boundary value problem, 𝐷₀^5/2₊𝑢(𝑡)+𝐹𝑡,𝑢(𝑡),𝑢^′(𝑡)𝑢(0)∋0,0<𝑡<+∞,=𝑢^′(0)=0,𝐷^3/2𝑢()−1+∞8𝑢18−14𝑢(1)=0. (61) Here 𝑚=4,𝛼=5/2, 𝛽₁=1/8,𝛽₂=1/4,𝜉₁=1/8, and 𝜉₂=1, and 𝐹∶𝐼×ℝ×ℝ→𝒫(ℝ) is a multivalued map given by 𝐹𝑡,𝑥,𝑦=𝑒^−𝑡1+𝑡^3/2𝑥+𝑦⁵𝑥+𝑦⁵,+5+92𝑒^−𝑡1+𝑡^3/2𝑥+𝑦³𝑥+𝑦³.+3+1 (62) For 𝑣∈𝐹, we have 𝑣(𝑡)1+𝑡^3/2≤max𝑒^−𝑡𝑥+𝑦⁵𝑥+𝑦⁵,+5+92𝑒^−𝑡𝑥+𝑦³𝑥+𝑦³+3+1≤10𝑒^−𝑡,𝑥,𝑦∈ℝ. (63) Thus, 𝐹(𝑡,𝑥,𝑦)_𝒫𝑣(𝑡)∶=sup1+𝑡^3/2∶𝑣∈𝐹𝑡,𝑥,𝑦≤10𝑒^−𝑡=𝜑(𝑡)𝜓(‖𝑥‖),𝑥,𝑦∈ℝ, (64) with 𝜑(𝑡)=𝑒^−𝑡, 𝜓(‖𝑥‖)=10.

Also, by direct calculation, we can obtain that 𝐿₁=1.01529 and 𝐿₂=3.045869. Further, by using the following condition: 𝑀max𝐿₁,𝐿₂𝜓(𝑀)∫₀^+∞𝜑(𝑠)𝑑𝑠>1, (65) we find that 𝑀>30.45869. Clearly, all the conditions of Theorem 21 are satisfied. So, there exists at least one solution of problem (1) on 𝐼.

Acknowledgments

The authors thank the referees for their careful reading of this paper and useful suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.