Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions

Abstract

We discuss the existence of solutions, under the Pettis integrability assumption, for a class of boundary value problems for fractional differential inclusions involving nonlinear nonseparated boundary conditions. Our analysis relies on the Mönch fixed point theorem combined with the technique of measures of weak noncompactness.

1. Introduction

This paper is mainly concerned with the existence results for the following fractional differential inclusion with non-separated boundary conditions: 𝑢(0)=𝜆𝐷^𝑐𝛼₁𝑢(𝑡)∈𝐹(𝑡,𝑢(𝑡)),𝑡∈𝐽∶=[0,𝑇],𝑇>0,𝑢(𝑇)+𝜇₁,𝑢′(0)=𝜆₂𝑢′(𝑇)+𝜇₂,𝜆₁≠1,𝜆₂≠1,(1.1) where 1<𝛼≤2 is a real number, ^𝑐𝐷^𝛼 is the Caputo fractional derivative. 𝐹∶𝐽×𝐸→𝒫(𝐸) is a multivalued map, 𝐸 is a Banach space with the norm ‖⋅‖, and 𝒫(𝐸) is the family of all nonempty subsets of 𝐸.

Recently, fractional differential equations have found numerous applications in various fields of physics and engineering [1, 2]. It should be noted that most of the books and papers on fractional calculus are devoted to the solvability of initial value problems for differential equations of fractional order. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quite recently and many aspects of this theory need to be explored. For more details and examples, see [3–18] and the references therein.

To investigate the existence of solutions of the problem above, we use Mönch’s fixed point theorem combined with the technique of measures of weak noncompactness, which is an important method for seeking solutions of differential equations. This technique was mainly initiated in the monograph of Banaś and Goebel [19] and subsequently developed and used in many papers; see, for example, Banaś and Sadarangani [20], Guo et al. [21], Krzyśka and Kubiaczyk [22], Lakshmikantham and Leela [23], Mönch’s [24], O’Regan [25, 26], Szufla [27, 28], and the references therein.

In 2007, Ouahab [29] investigated the existence of solutions for 𝛼-fractional differential inclusions by means of selection theorem together with a fixed point theorem. Very recently, Chang and Nieto [30] established some new existence results for fractional differential inclusions due to fixed point theorem of multivalued maps. Problem (1.1) was discussed for single valued case in the paper [31]; some existence results for single- and multivalued cases for an extension of (1.1) to non-separated integral boundary conditions were obtained in the article [32] and [33]. About other results on fractional differential inclusions, we refer the reader to [34]. As far as we know, there are very few results devoted to weak solutions of nonlinear fractional differential inclusions. Motivated by the above mentioned papers, the purpose of this paper is to establish the existence results for the boundary value problem (1.1) by virtue of the Mönch fixed point theorem combined with the technique of measures of weak noncompactness.

The remainder of this paper is organized as follows. In Section 2, we present some basic definitions and notations about fractional calculus and multivalued maps. In Section 3, we give main results for fractional differential inclusions. In the last section, an example is given to illustrate our main result.

2. Preliminaries and Lemmas

In this section, we introduce notation, definitions, and preliminary facts that will be used in the remainder of this paper. Let 𝐸 be a real Banach space with norm ‖⋅‖ and dual space 𝐸^∗, and let (𝐸,𝜔)=(𝐸,𝜎(𝐸,𝐸^∗)) denote the space 𝐸 with its weak topology. Here, let 𝐶(𝐽,𝐸) be the Banach space of all continuous functions from 𝐽 to 𝐸 with the norm ‖𝑦‖_∞=sup{‖𝑦(𝑡)‖∶0≤𝑡≤𝑇},(2.1) and let 𝐿¹(𝐽,𝐸) denote the Banach space of functions 𝑦∶𝐽→𝐸 that are the Lebesgue integrable with norm ‖𝑦‖_𝐿¹=^𝑇0‖𝑦(𝑡)‖𝑑𝑡.(2.2) We let 𝐿^∞(𝐽,𝐸) to be the Banach space of bounded measurable functions 𝑦∶𝐽→𝐸 equipped with the norm ‖𝑦‖_𝐿^∞=inf{𝑐>0∶‖𝑦(𝑡)‖≤𝑐,a.e.𝑡∈𝐽}.(2.3) Also, 𝐴𝐶¹(𝐽,𝐸) will denote the space of functions 𝑦∶𝐽→𝐸 that are absolutely continuous and whose first derivative, 𝑦^, is absolutely continuous.

Let (𝐸,‖⋅‖) be a Banach space, and let 𝑃_cl(𝐸)={𝑌∈𝒫(𝐸)∶𝑌isclosed}, 𝑃_𝑏(𝐸)={𝑌∈𝒫(𝐸)∶𝑌isbounded}, 𝑃_cp(𝐸)={𝑌∈𝒫(𝐸)∶𝑌iscompact}, and 𝑃_cp,𝑐(𝐸)={𝑌∈𝒫(𝐸)∶𝑌iscompactandconvex}. A multivalued map 𝐺∶𝐸→𝑃(𝐸) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥∈𝐸. We say that 𝐺 is bounded on bounded sets if 𝐺(𝐵)=∪_𝑥∈𝐵𝐺(𝑥) is bounded in 𝐸 for all 𝐵∈𝑃_𝑏(𝐸) (i.e., sup_𝑥∈𝐵{sup{‖𝑦‖∶𝑦∈𝐺(𝑥)}}<∞). The mapping 𝐺 is called upper semicontinuous (u.s.c.) on 𝐸 if for each 𝑥₀∈𝐸, the set 𝐺(𝑥₀) is a nonempty closed subset of 𝐸 and if for each open set 𝑁 of 𝐸 containing 𝐺(𝑥₀), there exists an open neighborhood 𝑁₀ of 𝑥₀ such that 𝐺(𝑁₀)⊆𝑁. We say that 𝐺 is completely continuous if 𝐺(ℬ) is relatively compact for every ℬ∈𝑃_𝑏(𝐸). If the multivalued map 𝐺 is completely continuous with nonempty compact values, then 𝐺 is u.s.c. if and only if 𝐺 has a closed graph (i.e., 𝑥_𝑛→𝑥_∗,𝑦_𝑛→𝑦_∗,𝑦_𝑛∈𝐺(𝑥_𝑛) imply 𝑦_∗∈𝐺(𝑥_∗)). The mapping 𝐺 has a fixed point if there is 𝑥∈𝐸 such that 𝑥∈𝐺(𝑥). The set of fixed points of the multivalued operator 𝐺 will be denoted by Fix𝐺. A multivalued map 𝐺∶𝐽→𝑃_cl(𝐸) is said to be measurable if for every 𝑦∈𝐸, the function 𝑡⟼𝑑(𝑦,𝐺(𝑡))=inf||𝑦−𝑧||∶𝑧∈𝐺(𝑡)(2.4) is measurable. For more details on multivalued maps, see the books of Aubin and Cellina [35], Aubin and Frankowska [36], Deimling [37], Hu and Papageorgiou [38], Kisielewicz [39], and Covitz and Nadler [40].

Moreover, for a given set 𝑉 of functions 𝑣∶𝐽↦ℝ, let us denote by 𝑉(𝑡)={𝑣(𝑡)∶𝑣∈𝑉}, 𝑡∈𝐽, and 𝑉(𝐽)={𝑣(𝑡)∶𝑣∈𝑉,𝑡∈𝐽}.

For any 𝑦∈𝐶(𝐽,𝐸), let 𝑆_𝐹,𝑦 be the set of selections of 𝐹 defined by 𝑆_𝐹,𝑦=𝑓∈𝐿¹(𝐽,𝐸)∶𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e.𝑡∈𝐽.(2.5)

Definition 2.1. A function ℎ∶𝐸→𝐸 is said to be weakly sequentially continuous if ℎ takes each weakly convergent sequence in 𝐸 to a weakly convergent sequence in 𝐸 (i.e., for any (𝑥_𝑛)_𝑛 in 𝐸 with 𝑥_𝑛(𝑡)→𝑥(𝑡) in (𝐸,𝜔) then ℎ(𝑥_𝑛(𝑡))→ℎ(𝑥(𝑡)) in (𝐸,𝜔) for each 𝑡→𝐽).

Definition 2.2. A function 𝐹∶𝑄→𝑃_cl,cv(𝑄) has a weakly sequentially closed graph if for any sequence (𝑥_𝑛,𝑦_𝑛)^∞1∈𝑄×𝑄, 𝑦_𝑛∈𝐹(𝑥_𝑛) for 𝑛∈{1,2,…} with 𝑥_𝑛(𝑡)→𝑥(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽 and 𝑦_𝑛(𝑡)→𝑦(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽, then 𝑦∈𝐹(𝑥).

Definition 2.3 (see [41]). The function 𝑥∶𝐽→𝐸 is said to be the Pettis integrable on 𝐽 if and only if there is an element 𝑥_𝐽∈𝐸 corresponding to each 𝐼⊂𝐽 such that 𝜑(𝑥_𝐼)=∫_𝐼𝜑(𝑥(𝑠))𝑑𝑠 for all 𝜑∈𝐸^∗, where the integral on the right is supposed to exist in the sense of Lebesgue. By definition, 𝑥_𝐼=∫_𝐼𝑥(𝑠)𝑑𝑠.
Let 𝑃(𝐽,𝐸) be the space of all 𝐸-valued Pettis integrable functions in the interval 𝐽.

Lemma 2.4 (see [41]). If 𝑥(⋅) is Pettis’ integrable and ℎ(⋅) is a measurable and essentially bounded real-valued function, then 𝑥(⋅)ℎ(⋅) is Pettis’ integrable.

Definition 2.5 (see [42]). Let 𝐸 be a Banach space, Ω_𝐸 the set of all bounded subsets of 𝐸, and 𝐵₁ the unit ball in 𝐸. The De Blasi measure of weak noncompactness is the map 𝛽∶Ω_𝐸→[0,∞) defined by 𝛽(𝑋)=inf𝜖>0∶thereexistsaweaklycompactsubsetΩof𝐸suchthat𝑋⊂𝜖𝐵₁+Ω.(2.6)

Lemma 2.6 (see [42]). The De Blasi measure of noncompactness satisfies the following properties:(a)𝑆⊂𝑇⇒𝛽(𝑆)≤𝛽(𝑇);(b)𝛽(𝑆)=0⇔𝑆 is relatively weakly compact;(c)𝛽(𝑆∪𝑇)=max{𝛽(𝑆),𝛽(𝑇)};(d)𝛽(𝑆^𝜔)=𝛽(𝑆), where 𝑆^𝜔 denotes the weak closure of 𝑆;(e)𝛽(𝑆+𝑇)≤𝛽(𝑆)+𝛽(𝑇);(f)𝛽(𝑎𝑆)=|𝑎|𝛼(𝑆);(g)𝛽(conv(𝑆))=𝛽(𝑆);(h)𝛽(∪_|𝜆|≤ℎ𝜆𝑆)=ℎ𝛽(𝑆).
The following result follows directly from the Hahn-Banach theorem.

Lemma 2.7. Let 𝐸 be a normed space with 𝑥₀≠0. Then there exists 𝜑∈𝐸^∗ with ‖𝜑‖=1 and 𝜑(𝑥₀)=‖𝑥₀‖.
For completeness, we recall the definitions of the Pettis-integral and the Caputo derivative of fractional order.

Definition 2.8 (see [25]). Let ℎ∶𝐽→𝐸 be a function. The fractional Pettis integral of the function ℎ of order 𝛼∈ℝ⁺ is defined by 𝐼^𝛼ℎ(𝑡)=^𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)ℎ(𝑠)𝑑𝑠,(2.7) where the sign “∫” denotes the Pettis integral and Γ is the gamma function.

Definition 2.9 (see [3]). For a function ℎ∶𝐽→𝐸, the Caputo fractional-order derivative of ℎ is defined by ^𝑐𝐷^𝛼𝑎+ℎ(𝑡)=Γ(𝑛1−𝛼)^𝑡𝑎(𝑡−𝑠)^{𝑛−𝛼−1}ℎ^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝛼<𝑛,(2.8) where 𝑛=[𝛼]+1 and [𝛼] denotes the integer part of 𝛼.

Lemma 2.10 (see [43]). Let 𝐸 be a Banach space with 𝑄 a nonempty, bounded, closed, convex, equicontinuous subset of 𝐶(𝐽,𝐸). Suppose 𝐹∶𝑄→𝑃_cl,cv(𝑄) has a weakly sequentially closed graph. If the implication 𝑉=conv({0}∪𝐹(𝑉))⟹𝑉𝑖𝑠𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦𝑤𝑒𝑎𝑘𝑙𝑦𝑐𝑜𝑚𝑝𝑎𝑐𝑡(2.9) holds for every subset 𝑉 of 𝑄, then the operator inclusion 𝑥∈𝐹(𝑥) has a solution in 𝑄.

3. Main Results

Let us start by defining what we mean by a solution of problem (1.1).

Definition 3.1. A function 𝑦∈𝐴𝐶¹(𝐽,𝐸) is said to be a solution of (1.1), if there exists a function 𝑣∈𝐿¹(𝐽,𝐸) with 𝑣(𝑡)∈𝐹(𝑡,𝑦(𝑡)) for a.e. 𝑡∈𝐽, such that ^𝑐𝐷^𝛼𝑦(𝑡)=𝑣(𝑡)a.e.𝑡∈𝐽,1<𝛼≤2,(3.1) and 𝑦 satisfies conditions 𝑢(0)=𝜆₁𝑢(𝑇)+𝜇₁,𝑢′(0)=𝜆₂𝑢′(𝑇)+𝜇₂,𝜆₁≠1,𝜆₂≠1.
To prove the main results, we need the following assumptions:(H1)𝐹∶𝐽×𝐸→𝑃_cp,cv(𝐸) has weakly sequentially closed graph;(H2)for each continuous 𝑥∈𝐶(𝐽,𝐸), there exists a scalarly measurable function 𝑣∶𝐽→𝐸 with 𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)) a.e. on 𝐽 and 𝑣 is Pettis integrable on 𝐽;(H3)there exist 𝑝_𝑓∈𝐿^∞(𝐽,ℝ⁺) and a continuous nondecreasing function 𝜓∶[0,∞)→[0,∞) such that ‖𝐹(𝑡,𝑢)‖=sup{|𝑣|∶𝑣∈𝐹(𝑡,𝑢)}≤𝑝_𝑓(𝑡)𝜓(‖𝑢‖);(3.2)(H4)for each bounded set 𝐷⊂𝐸, and each 𝑡∈𝐼, the following inequality holds: 𝛽(𝐹(𝑡,𝐷))≤𝑝_𝑓(𝑡)⋅𝛽(𝐷);(3.3)(H5)there exists a constant 𝑅>0 such that 𝑅𝑔^∗+‖‖𝑝_𝑓‖‖_𝐿∞𝜓(𝑅)𝐺^∗>1,(3.4)where 𝑔^∗ and 𝐺^∗ are defined by (3.9).

Theorem 3.2. Let 𝐸 be a Banach space. Assume that hypotheses (H1)–(H5) are satisfied. If ‖‖𝑝_𝑓‖‖_𝐿∞𝐺^∗<1,(3.5) then the problem (1.1) has at least one solution on 𝐽.

Proof. Let 𝜌∈𝐶[0,𝑇] be a given function; it is obvious that the boundary value problem [18] 𝑢(𝑡)=𝜆₁𝐷^𝑐𝛼𝑢(𝑡)=𝜌(𝑡),𝑡∈(0,𝑇),1<𝛼≤2𝑢(𝑇)+𝜇₁,𝑢^(0)=𝜆₂𝑢^(𝑇)+𝜇₂,𝜆₁≠1,𝜆₂≠1(3.6) has a unique solution 𝑢(𝑡)=^𝑇0𝐺(𝑡,𝑠)𝜌(𝑠)𝑑𝑠+𝑔(𝑡),(3.7) where 𝐺(𝑡,𝑠) is defined by the formula 𝐺(𝑡,𝑠)=⎧⎪⎪⎨⎪⎪⎩(𝑡−𝑠)^𝛼−1Γ(𝛼)−𝜆¹(𝑇−𝑠)^𝛼−1𝜆₁−1Γ(𝛼)+𝜆²𝜆₁𝑇+1−𝜆₁𝑡(𝑇−𝑠)^𝛼−2𝜆₂−1𝜆₁−1Γ(𝛼−1),−𝜆if0≤𝑠≤𝑡≤𝑇,₁(𝑇−𝑠)^𝛼−1𝜆₁−1Γ(𝛼)+𝜆²𝜆₁𝑇+1−𝜆₁𝑡(𝑇−𝑠)^𝛼−2𝜆₂−1𝜆₁−1Γ(𝛼−1),𝑔(𝑡)=𝜇₂if0≤𝑡≤𝑠≤𝑇,𝜆₁𝑇+1−𝜆₁𝑡𝜆₂−1𝜆₁−1−𝜆₁𝜇−¹1.(3.8)
From the expression of 𝐺(𝑡,𝑠) and 𝑔(𝑡), it is obvious that 𝐺(𝑡,𝑠) is continuous on 𝐽×𝐽 and 𝑔(𝑡) is continuous on 𝐽. Denote by 𝐺^∗=sup^𝑇0||𝐺(𝑡,𝑠)||𝑑𝑠,𝑡∈𝐽,𝑔^∗=max_{0≤𝑡≤𝑇}‖𝑔(𝑡)‖.(3.9)
We transform the problem (1.1) into fixed point problem by considering the multivalued operator 𝑁∶𝐶(𝐽,𝐸)→𝑃_cl,cv(𝐶(𝐽,𝐸)) defined by 𝑁(𝑥)=ℎ∈𝐶(𝐽,𝐸)∶ℎ(𝑡)=𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠,𝑣∈𝑆_𝐹,𝑥,(3.10) and refer to [31] for defining the operator 𝑁. Clearly, the fixed points of 𝑁 are solutions of Problem (1.1). We first show that (3.10) makes sense. To see this, let 𝑥∈𝐶(𝐽,𝐸); by (H2) there exists a Pettis’ integrable function 𝑣∶𝐽→𝐸 such that 𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)) for a.e. 𝑡∈𝐽. Since 𝐺(𝑡,⋅)∈𝐿^∞(𝐽), then 𝐺(𝑡,⋅)𝑣(⋅) is Pettis integrable and thus 𝑁 is well defined.
Let 𝑅>0, and consider the set 𝐷=+𝑥‖‖∈𝑝_𝑓𝐶(𝐽,𝐸)∶‖𝑥‖_∞≤𝑅,‖‖𝑥𝑡₁−𝑥𝑡₂‖‖≤‖‖𝑔𝑡₁−𝑔𝑡₂‖‖‖‖_𝐿∞𝜓(𝑅)^𝑇0‖‖𝐺𝑡₂,𝑠−𝐺𝑡₁,𝑠‖‖𝑑𝑠for𝑡₁,𝑡₂∈𝐽;(3.11) clearly, the subset 𝐷 is a closed, convex, bounded, and equicontinuous subset of 𝐶(𝐽,𝐸). We shall show that 𝑁 satisfies the assumptions of Lemma 2.10. The proof will be given in four steps.
Step  1. We will show that the operator 𝑁(𝑥) is convex for each 𝑥∈𝐷.
Indeed, if ℎ₁ and ℎ₂ belong to 𝑁(𝑥), then there exists Pettis’ integrable functions 𝑣₁(𝑡), 𝑣₂(𝑡)∈𝐹(𝑡,𝑥(𝑡)) such that, for all 𝑡∈𝐽, we have ℎ_𝑖(𝑡)=𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣_𝑖(𝑠)𝑑𝑠,𝑖=1,2.(3.12) Let 0≤𝑑≤1. Then, for each 𝑡∈𝐽, we have 𝑑ℎ₁+(1−𝑑)ℎ₂(𝑡)=𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑑𝑣₁(𝑠)+(1−𝑑)𝑣₂(𝑠)𝑑𝑠.(3.13) Since 𝐹 has convex values, (𝑑𝑣₁+(1−𝑑)𝑣₂)(𝑡)∈𝐹(𝑡,𝑦) and we have 𝑑ℎ₁+(1−𝑑)ℎ₂∈𝑁(𝑥).
Step  2. We will show that the operator 𝑁 maps 𝐷 into 𝐷.
To see this, take 𝑢∈𝑁𝐷. Then there exists 𝑥∈𝐷 with 𝑢∈𝑁(𝑥) and there exists a Pettis integrable function 𝑣∶𝐽→𝐸 with 𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)) for a.e. 𝑡∈𝐽. Without loss of generality, we assume 𝑢(𝑠)≠0 for all 𝑠∈𝐽. Then, there exists 𝜑_𝑠∈𝐸^∗ with ‖𝜑_𝑠‖=1 and 𝜑_𝑠(𝑢(𝑠))=‖𝑢(𝑠)‖. Hence, for each fixed 𝑡∈𝐽, we have ‖𝑢(𝑡)‖=𝜑_𝑡(𝑢(𝑡))=𝜑_𝑡𝑔(𝑡)+^𝑇0≤𝜑_𝑡𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠(𝑔(𝑡))+𝜑_𝑡^𝑇0≤‖𝑔(𝑡)‖+_𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠≤𝑔_∗‖𝐺(𝑡,𝑠)‖𝜑_𝑡(𝑣(𝑠))𝑑𝑠+𝐺^∗𝜓‖𝑥‖_∞‖‖𝑝_𝑓‖‖_𝐿∞.(3.14)
Therefore, by (H5), we have ‖𝑢‖_∞≤𝑔^∗+‖‖𝑝_𝑓‖‖_𝐿∞𝐺^∗𝜓‖𝑅‖_∞≤𝑅.(3.15)
Next suppose 𝑢∈𝑁𝐷 and 𝜏₁,𝜏₂∈𝐽, with 𝜏₁<𝜏₂ so that 𝑢(𝜏₂)−𝑢(𝜏₁)≠0. Then, there exists 𝜑∈𝐸^∗ such that ‖𝑢(𝜏₂)−𝑢(𝜏₁)‖=𝜑(𝑢(𝜏₂)−𝑢(𝜏₁)). Hence, ‖‖𝑢𝜏₂−𝑢𝜏₁‖‖≤=𝜑𝜑𝑔𝑔𝑡𝑡₂₂−𝑔𝑡₁+^𝑇0𝐺𝜏₂,𝑠−𝐺𝜏₁,𝑠⋅𝑣(𝑠)𝑑𝑠≤‖‖𝑔𝑡−𝑔𝑡₁+𝜑^𝑇0𝐺𝜏₂,𝑠−𝐺𝜏₁,𝑠⋅𝑣(𝑠)𝑑𝑠₂≤‖‖𝑔𝑡−𝑔𝑡₁‖‖+^𝑇0‖‖𝐺𝜏₂,𝑠−𝐺𝜏₁,𝑠‖⋅‖𝑣(𝑠)‖‖𝑑𝑠₂−𝑔𝑡₁‖‖+𝜓(𝑅)‖‖𝑝_𝑓‖‖_𝐿∞^𝑇0‖‖𝐺𝜏₂,𝑠−𝐺𝜏₁,𝑠‖‖𝑑𝑠;(3.16) this means that 𝑢∈𝐷.
Step  3. We will show that the operator 𝑁 has a weakly sequentially closed graph.
Let (𝑥_𝑛,𝑦_𝑛)^∞1 be a sequence in 𝐷×𝐷 with 𝑥_𝑛(𝑡)→𝑥(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽, 𝑦_𝑛(𝑡)→𝑦(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽, and 𝑦_𝑛∈𝑁(𝑥_𝑛) for 𝑛∈{1,2,…}. We will show that 𝑦∈𝑁𝑥. By the relation 𝑦_𝑛∈𝑁(𝑥_𝑛), we mean that there exists 𝑣_𝑛∈𝑆_{𝐹,𝑥𝑛} such that 𝑦_𝑛(𝑡)=𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣_𝑛(𝑠)𝑑𝑠.(3.17)
We must show that there exists 𝑣∈𝑆_𝐹,𝑥 such that, for each 𝑡∈𝐽, 𝑦(𝑡)=𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠.(3.18)
Since 𝐹(⋅,⋅) has compact values, there exists a subsequence 𝑣_𝑛𝑚 such that 𝑣_𝑛𝑚(𝑣⋅)⟶𝑣(⋅)in(𝐸,𝜔)as𝑚⟶∞_𝑛𝑚(𝑡)∈𝐹𝑡,𝑥_𝑛(𝑡)a.e.𝑡∈𝐽.(3.19) Since 𝐹(𝑡,⋅) has a weakly sequentially closed graph, 𝑣∈𝐹(𝑡,𝑥). The Lebesgue dominated convergence theorem for the Pettis integral then implies that for each 𝜑∈𝐸^∗, 𝜑𝑦_𝑛(𝑡)=𝜑𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣_𝑛(𝑠)𝑑𝑠⟶𝜑𝑔(𝑡)+^𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠;(3.20) that is, 𝑦_𝑛(𝑡)→𝑁𝑥(𝑡) in (𝐸,𝑤). Repeating this for each 𝑡∈𝐽 shows 𝑦(𝑡)∈𝑁𝑥(𝑡).
Step  4. The implication (2.9) holds. Now let 𝑉 be a subset of 𝐷 such that 𝑉⊂conv(𝑁(𝑉)∪{0}). Clearly, 𝑉(𝑡)⊂conv(𝑁(𝑉)∪{0}) for all 𝑡∈𝐽. Hence, 𝑁𝑉(𝑡)⊂𝑁𝐷(𝑡),𝑡∈𝐽, is bounded in 𝑃(𝐸).
Since function 𝑔 is continuous on 𝐽, the set {𝑔(𝑡),𝑡∈𝐽}⊂𝐸 is compact, so 𝛽(𝑔(𝑡))=0. By assumption (H4) and the properties of the measure 𝛽, we have for each 𝑡∈𝐽𝛽(𝑁(𝑉)(𝑡))=≤𝛽𝛽{𝑔𝑔(𝑡()𝑡)∶+𝑡∈^𝑇0𝐽𝐺}(+𝑡,𝑠𝛽)𝑣(𝑠_𝑇0)𝑑𝑠∶𝑣∈𝑆_𝐹,𝑥,𝑥∈𝑉,𝑡∈𝐽≤𝛽_𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠∶𝑣∈𝑆_𝐹,𝑥,𝑥∈𝑉,𝑡∈𝐽≤𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠∶𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)),𝑥∈𝑉,𝑡∈𝐽_𝑇0≤‖‖𝑝‖𝐺(𝑡,𝑠)‖⋅𝑝_𝑓(𝑠)⋅𝛽(𝑉(𝑠))𝑑𝑠_𝑓≤‖‖𝑝‖‖_𝐿∞⋅^𝑇0‖𝐺(𝑡,𝑠)‖⋅𝛽(𝑉(𝑠))𝑑𝑠_𝑓‖‖_𝐿∞⋅𝐺^∗⋅^𝑇0𝛽(𝑉(𝑠))𝑑𝑠,(3.21) which gives ‖𝑣‖_∞≤‖‖𝑝_𝑓‖‖_𝐿∞⋅‖𝑣‖_∞⋅𝐺^∗.(3.22)
This means that ‖𝑣‖_∞⋅1−‖‖𝑝_𝑓‖‖_𝐿∞⋅𝐺^∗≤0.(3.23) By (3.5) it follows that ‖𝑣‖_∞=0; that is, 𝑣=0 for each 𝑡∈𝐽, and then 𝑉 is relatively weakly compact in 𝐸. In view of Lemma 2.10, we deduce that 𝑁 has a fixed point which is obviously a solution of Problem (1.1). This completes the proof.

In the sequel we present an example which illustrates Theorem 3.2.

4. An Example

Example 4.1. We consider the following partial hyperbolic fractional differential inclusion of the form ^𝑐𝐷^𝛼𝑢_𝑛(𝑡)∈17𝑒^𝑡+13𝑢(0)=𝜆₁1+||𝑢_𝑛(𝑡)||,𝑡∈𝐽∶=[0,𝑇],1<𝛼≤2,𝑢(𝑇)+𝜇₁,𝑢^(0)=𝜆₂𝑢^(𝑇)+𝜇₂,(4.1)
Set 𝑇=1, 𝜆₁=𝜆₂=−1, 𝜇₁=𝜇₂=0, then 𝑔(𝑡)=0. So 𝑔^∗=0.
Let 𝐸=𝑙¹=𝑢=𝑢₁,𝑢₂,…,𝑢_𝑛,…∶^∞_𝑛=1||𝑢_𝑛||<∞(4.2) with the norm ‖𝑢‖_𝐸=^∞_𝑛=1||𝑢_𝑛||.(4.3) Set 𝑢=𝑢𝑓₁,𝑢₂,…,𝑢_𝑛,…,𝑓=𝑓₁,𝑓₂,…,𝑓_𝑛,…_𝑛,𝑡,𝑢_𝑛=17𝑒^𝑡+131+||𝑢_𝑛||,𝑡∈𝐽.(4.4) For each 𝑢_𝑛∈ℝ and 𝑡∈𝐽, we have ||𝑓_𝑛𝑡,𝑢_𝑛||≤17𝑒^𝑡+131+||𝑢_𝑛||.(4.5)
Hence conditions (H1), (H2), and (H3) hold with 𝑝_𝑓(𝑡)=1/(7𝑒^𝑡+13),𝑡∈𝐽, and 𝜓(𝑢)=1+𝑢,𝑢∈[0,∞). For any bounded set 𝐷⊂𝑙¹, we have 𝛽(𝐹(𝑡,𝐷))≤17𝑒^𝑡+13⋅𝛽(𝐷),∀𝑡∈𝐽.(4.6) Hence (H4) is satisfied. From (3.8), we have 𝐺(𝑡,𝑠)=⎧⎪⎨⎪⎩(𝑡−Γ(𝑠𝛼))^𝛼−1−(12−Γ𝑠(𝛼))^𝛼−1+(1−42Γ𝑡)(𝛼(1−−1𝑠))^𝛼−2,−(1−𝑠)_𝛼−1if0≤𝑠≤𝑡≤1,2Γ(𝛼)+(1−42Γ𝑡)(𝛼(1−−1𝑠))^𝛼−2,if0≤𝑡≤𝑠≤1.(4.7) So, we get ¹⁰𝐺(𝑡,𝑠)𝑑𝑠==^𝑡0𝐺(𝑡,𝑠)𝑑𝑠+^1𝑡𝐺(𝑡,𝑠)𝑑𝑠_𝑡0+(𝑡−Γ(𝑠𝛼))^𝛼−1−(12−Γ𝑠(𝛼))^𝛼−1+(1−42Γ𝑡)(𝛼(1−−1𝑠))^𝛼−2𝑑𝑠_1𝑡=4𝑡_𝛼−(12−Γ𝑠(𝛼))^𝛼−1+(1−42Γ𝑡)(𝛼(1−−1𝑠))^𝛼−2𝑑𝑠−24Γ(𝛼+1)+41Γ−(𝛼2𝑡).(4.8) A simple computation gives 𝐺^∗<14Γ(𝛼)+2Γ(𝛼1+1)∶=𝐴^𝛼.(4.9) We shall check that condition (3.5) is satisfied. Indeed ‖𝑝‖_𝐿^∞𝐺^∗<17𝑒¹³𝐴_𝛼<1,(4.10) which is satisfied for some 𝛼∈(1,2], and (H5) is satisfied for 𝑅>𝐴_𝛼/(7𝑒¹³−𝐴_𝛼). Then by Theorem 3.2, the problem (4.1) has at least one solution on 𝐽 for values of 𝛼 satisfying (4.10).

Acknowledgments

The first author’s work was supported by NNSF of China (11161027), NNSF of China (10901075), and the Key Project of Chinese Ministry of Education (210226). The authors are grateful to the referees for their comments according to which the paper has been revised.