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Journal of Applied Mathematics
Volume 2012, Article ID 530624, 13 pages
http://dx.doi.org/10.1155/2012/530624
Research Article

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

1Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China
2College of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received 24 April 2012; Revised 21 July 2012; Accepted 22 July 2012

Academic Editor: Ram N. Mohapatra

Copyright © 2012 Wen-Xue Zhou and Hai-Zhong Liu. 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.

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: 𝑐𝐷𝛼[]𝑢(𝑡)𝐹(𝑡,𝑢(𝑡)),𝑡𝐽=0,𝑇,𝑇>0,𝑢(0)=𝜆1𝑢(𝑇)+𝜇1,𝑢(0)=𝜆2𝑢(𝑇)+𝜇2,𝜆11,𝜆21,(1.1) where 1<𝛼2 is a real number, 𝑐𝐷𝛼 is the Caputo fractional derivative. 𝐹𝐽×𝐸𝒫(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 [318] 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)0𝑡𝑇},(2.1) and let 𝐿1(𝐽,𝐸) denote the Banach space of functions 𝑦𝐽𝐸 that are the Lebesgue integrable with norm 𝑦𝐿1=𝑇0𝑦(𝑡)𝑑𝑡.(2.2) We let 𝐿(𝐽,𝐸) to be the Banach space of bounded measurable functions 𝑦𝐽𝐸 equipped with the norm 𝑦𝐿=inf{𝑐>0𝑦(𝑡)𝑐,a.e.𝑡𝐽}.(2.3) Also, 𝐴𝐶1(𝐽,𝐸) will denote the space of functions 𝑦𝐽𝐸 that are absolutely continuous and whose first derivative, 𝑦, is absolutely continuous.

Let (𝐸,) be a Banach space, and let 𝑃cl(𝐸)={𝑌𝒫(𝐸)𝑌isclosed}, 𝑃𝑏(𝐸)={𝑌𝒫(𝐸)𝑌isbounded}, 𝑃cp(𝐸)={𝑌𝒫(𝐸)𝑌iscompact}, and 𝑃cp,𝑐(𝐸)={𝑌𝒫(𝐸)𝑌iscompactandconvex}. A multivalued map 𝐺𝐸𝑃(𝐸) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥𝐸. We say that 𝐺 is bounded on bounded sets if 𝐺(𝐵)=𝑥𝐵𝐺(𝑥) is bounded in 𝐸 for all 𝐵𝑃𝑏(𝐸) (i.e., sup𝑥𝐵{sup{𝑦𝑦𝐺(𝑥)}}<). The mapping 𝐺 is called upper semicontinuous (u.s.c.) on 𝐸 if for each 𝑥0𝐸, 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() is relatively compact for every 𝑃𝑏(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., 𝑥𝑛𝑥,𝑦𝑛𝑦,𝑦𝑛𝐺(𝑥𝑛) imply 𝑦𝐺(𝑥)). The mapping 𝐺 has a fixed point if there is 𝑥𝐸 such that 𝑥𝐺(𝑥). The set of fixed points of the multivalued operator 𝐺 will be denoted by Fix𝐺. A multivalued map 𝐺𝐽𝑃cl(𝐸) is said to be measurable if for every 𝑦𝐸, the function ||||𝑡𝑑(𝑦,𝐺(𝑡))=inf𝑦𝑧𝑧𝐺(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 𝑣𝐽, let us denote by 𝑉(𝑡)={𝑣(𝑡)𝑣𝑉}, 𝑡𝐽, and 𝑉(𝐽)={𝑣(𝑡)𝑣𝑉,𝑡𝐽}.

For any 𝑦𝐶(𝐽,𝐸), let 𝑆𝐹,𝑦 be the set of selections of 𝐹 defined by 𝑆𝐹,𝑦=𝑓𝐿1(𝐽,𝐸)𝑓(𝑡)𝐹(𝑡,𝑦(𝑡))a.e..𝑡𝐽(2.5)

Definition 2.1. A function 𝐸𝐸 is said to be weakly sequentially continuous if takes each weakly convergent sequence in 𝐸 to a weakly convergent sequence in 𝐸 (i.e., for any (𝑥𝑛)𝑛 in 𝐸 with 𝑥𝑛(𝑡)𝑥(𝑡) in (𝐸,𝜔) then (𝑥𝑛(𝑡))(𝑥(𝑡)) in (𝐸,𝜔) for each 𝑡𝐽).

Definition 2.2. A function 𝐹𝑄𝑃cl,cv(𝑄) has a weakly sequentially closed graph if for any sequence (𝑥𝑛,𝑦𝑛)1𝑄×𝑄, 𝑦𝑛𝐹(𝑥𝑛) for 𝑛{1,2,} with 𝑥𝑛(𝑡)𝑥(𝑡) in (𝐸,𝜔) for each 𝑡𝐽 and 𝑦𝑛(𝑡)𝑦(𝑡) in (𝐸,𝜔) for each 𝑡𝐽, then 𝑦𝐹(𝑥).

Definition 2.3 (see [41]). The function 𝑥𝐽𝐸 is said to be the Pettis integrable on 𝐽 if and only if there is an element 𝑥𝐽𝐸 corresponding to each 𝐼𝐽 such that 𝜑(𝑥𝐼)=𝐼𝜑(𝑥(𝑠))𝑑𝑠 for all 𝜑𝐸, where the integral on the right is supposed to exist in the sense of Lebesgue. By definition, 𝑥𝐼=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 () is a measurable and essentially bounded real-valued function, then 𝑥()() is Pettis’ integrable.

Definition 2.5 (see [42]). Let 𝐸 be a Banach space, Ω𝐸 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 𝛽Ω𝐸[0,) defined by 𝛽(𝑋)=inf𝜖>0thereexistsaweaklycompactsubsetΩof𝐸suchthat𝑋𝜖𝐵1.+Ω(2.6)

Lemma 2.6 (see [42]). The De Blasi measure of noncompactness satisfies the following properties:(a)𝑆𝑇𝛽(𝑆)𝛽(𝑇);(b)𝛽(𝑆)=0𝑆 is relatively weakly compact;(c)𝛽(𝑆𝑇)=max{𝛽(𝑆),𝛽(𝑇)};(d)𝛽(𝑆𝜔)=𝛽(𝑆), where 𝑆𝜔 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|𝜆𝑆)=𝛽(𝑆).
The following result follows directly from the Hahn-Banach theorem.

Lemma 2.7. Let 𝐸 be a normed space with 𝑥00. Then there exists 𝜑𝐸 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 𝐽𝐸 be a function. The fractional Pettis integral of the function of order 𝛼+ is defined by 𝐼𝛼(𝑡)=𝑡0(𝑡𝑠)𝛼1Γ(𝛼)(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 𝐽𝐸, the Caputo fractional-order derivative of is defined by 𝑐𝐷𝛼𝑎+(1𝑡)=Γ(𝑛𝛼)𝑡𝑎(𝑡𝑠)𝑛𝛼1(𝑛)(𝑠)𝑑𝑠,𝑛1<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 𝐹𝑄𝑃cl,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 𝑥𝐹(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 𝑦𝐴𝐶1(𝐽,𝐸) is said to be a solution of (1.1), if there exists a function 𝑣𝐿1(𝐽,𝐸) with 𝑣(𝑡)𝐹(𝑡,𝑦(𝑡)) for a.e. 𝑡𝐽, such that 𝑐𝐷𝛼𝑦(𝑡)=𝑣(𝑡)a.e.𝑡𝐽,1<𝛼2,(3.1) and 𝑦 satisfies conditions 𝑢(0)=𝜆1𝑢(𝑇)+𝜇1,𝑢(0)=𝜆2𝑢(𝑇)+𝜇2,𝜆11,𝜆21.
To prove the main results, we need the following assumptions:(H1)𝐹𝐽×𝐸𝑃cp,cv(𝐸) has weakly sequentially closed graph;(H2)for each continuous 𝑥𝐶(𝐽,𝐸), there exists a scalarly measurable function 𝑣𝐽𝐸 with 𝑣(𝑡)𝐹(𝑡,𝑥(𝑡)) a.e. on 𝐽 and 𝑣 is Pettis integrable on 𝐽;(H3)there exist 𝑝𝑓𝐿(𝐽,+) and a continuous nondecreasing function 𝜓[0,)[0,) such that 𝐹(𝑡,𝑢)=sup{|𝑣|𝑣𝐹(𝑡,𝑢)}𝑝𝑓(𝑡)𝜓(𝑢);(3.2)(H4)for each bounded set 𝐷𝐸, and each 𝑡𝐼, the following inequality holds: 𝛽(𝐹(𝑡,𝐷))𝑝𝑓(𝑡)𝛽(𝐷);(3.3)(H5)there exists a constant 𝑅>0 such that 𝑅𝑔+𝑝𝑓𝐿𝜓(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 𝑝𝑓𝐿𝐺<1,(3.5) then the problem (1.1) has at least one solution on 𝐽.

Proof. Let 𝜌𝐶[0,𝑇] be a given function; it is obvious that the boundary value problem [18] 𝑐𝐷𝛼𝑢(𝑡)=𝜌(𝑡),𝑡(0,𝑇),1<𝛼2𝑢(𝑡)=𝜆1𝑢(𝑇)+𝜇1,𝑢(0)=𝜆2𝑢(𝑇)+𝜇2,𝜆11,𝜆21(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)=(𝑡𝑠)𝛼1𝜆Γ(𝛼)1(𝑇𝑠)𝛼1𝜆1+𝜆1Γ(𝛼)2𝜆1𝑇+1𝜆1𝑡(𝑇𝑠)𝛼2𝜆2𝜆11,1Γ(𝛼1)if𝜆0𝑠𝑡𝑇,1(𝑇𝑠)𝛼1𝜆1+𝜆1Γ(𝛼)2𝜆1𝑇+1𝜆1𝑡(𝑇𝑠)𝛼2𝜆2𝜆11,1Γ(𝛼1)if𝜇0𝑡𝑠𝑇,𝑔(𝑡)=2𝜆1𝑇+1𝜆1𝑡𝜆2𝜆11𝜇11𝜆1.1(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=sup𝑇0||||𝐺(𝑡,𝑠)𝑑𝑠,𝑡𝐽,𝑔=max0𝑡𝑇𝑔(𝑡).(3.9)
We transform the problem (1.1) into fixed point problem by considering the multivalued operator 𝑁𝐶(𝐽,𝐸)𝑃cl,cv(𝐶(𝐽,𝐸)) defined by 𝑁(𝑥)=𝐶(𝐽,𝐸)(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠,𝑣𝑆𝐹,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 𝑥𝐶(𝐽,𝐸); by (H2) there exists a Pettis’ integrable function 𝑣𝐽𝐸 such that 𝑣(𝑡)𝐹(𝑡,𝑥(𝑡)) for a.e. 𝑡𝐽. 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=𝑥𝐶(𝐽,𝐸)𝑥𝑥𝑡𝑅,1𝑡𝑥2𝑔𝑡1𝑡𝑔2+𝑝𝑓𝐿𝜓(𝑅)𝑇0𝐺𝑡2𝑡,𝑠𝐺1,𝑠𝑑𝑠for𝑡1,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 𝑥𝐷.
Indeed, if 1 and 2 belong to 𝑁(𝑥), then there exists Pettis’ integrable functions 𝑣1(𝑡), 𝑣2(𝑡)𝐹(𝑡,𝑥(𝑡)) such that, for all 𝑡𝐽, we have 𝑖(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑖(𝑠)𝑑𝑠,𝑖=1,2.(3.12) Let 0𝑑1. Then, for each 𝑡𝐽, we have 𝑑1+(1𝑑)2(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑑𝑣1(𝑠)+(1𝑑)𝑣2(𝑠)𝑑𝑠.(3.13) Since 𝐹 has convex values, (𝑑𝑣1+(1𝑑)𝑣2)(𝑡)𝐹(𝑡,𝑦) and we have 𝑑1+(1𝑑)2𝑁(𝑥).
Step  2. We will show that the operator 𝑁 maps 𝐷 into 𝐷.
To see this, take 𝑢𝑁𝐷. Then there exists 𝑥𝐷 with 𝑢𝑁(𝑥) and there exists a Pettis integrable function 𝑣𝐽𝐸 with 𝑣(𝑡)𝐹(𝑡,𝑥(𝑡)) for a.e. 𝑡𝐽. Without loss of generality, we assume 𝑢(𝑠)0 for all 𝑠𝐽. Then, there exists 𝜑𝑠𝐸 with 𝜑𝑠=1 and 𝜑𝑠(𝑢(𝑠))=𝑢(𝑠). Hence, for each fixed 𝑡𝐽, we have 𝑢(𝑡)=𝜑𝑡(𝑢(𝑡))=𝜑𝑡𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠𝜑𝑡(𝑔(𝑡))+𝜑𝑡𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝜑𝑡(𝑣(𝑠))𝑑𝑠𝑔+𝐺𝜓𝑥𝑝𝑓𝐿.(3.14)
Therefore, by (H5), we have 𝑢𝑔+𝑝𝑓𝐿𝐺𝜓𝑅𝑅.(3.15)
Next suppose 𝑢𝑁𝐷 and 𝜏1,𝜏2𝐽, with 𝜏1<𝜏2 so that 𝑢(𝜏2)𝑢(𝜏1)0. Then, there exists 𝜑𝐸 such that 𝑢(𝜏2)𝑢(𝜏1)=𝜑(𝑢(𝜏2)𝑢(𝜏1)). Hence, 𝑢𝜏2𝜏𝑢1𝑔𝑡=𝜑2𝑡𝑔1+𝑇0𝐺𝜏2𝜏,𝑠𝐺1𝑔𝑡,𝑠𝑣(𝑠)𝑑𝑠𝜑2𝑡𝑔1+𝜑𝑇0𝐺𝜏2𝜏,𝑠𝐺1𝑔𝑡,𝑠𝑣(𝑠)𝑑𝑠2𝑡𝑔1+𝑇0𝐺𝜏2𝜏,𝑠𝐺1𝑔𝑡,𝑠𝑣(𝑠)𝑑𝑠2𝑡𝑔1𝑝+𝜓(𝑅)𝑓𝐿𝑇0𝐺𝜏2𝜏,𝑠𝐺1,𝑠𝑑𝑠;(3.16) this means that 𝑢𝐷.
Step  3. We will show that the operator 𝑁 has a weakly sequentially closed graph.
Let (𝑥𝑛,𝑦𝑛)1 be a sequence in 𝐷×𝐷 with 𝑥𝑛(𝑡)𝑥(𝑡) in (𝐸,𝜔) for each 𝑡𝐽, 𝑦𝑛(𝑡)𝑦(𝑡) in (𝐸,𝜔) for each 𝑡𝐽, and 𝑦𝑛𝑁(𝑥𝑛) for 𝑛{1,2,}. We will show that 𝑦𝑁𝑥. By the relation 𝑦𝑛𝑁(𝑥𝑛), we mean that there exists 𝑣𝑛𝑆𝐹,𝑥𝑛 such that 𝑦𝑛(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑛(𝑠)𝑑𝑠.(3.17)
We must show that there exists 𝑣𝑆𝐹,𝑥 such that, for each 𝑡𝐽, 𝑦(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠.(3.18)
Since 𝐹(,) has compact values, there exists a subsequence 𝑣𝑛𝑚 such that 𝑣𝑛𝑚()𝑣()in(𝐸,𝜔)as𝑣𝑚𝑛𝑚(𝑡)𝐹𝑡,𝑥𝑛(𝑡)a.e.𝑡𝐽.(3.19) Since 𝐹(𝑡,) has a weakly sequentially closed graph, 𝑣𝐹(𝑡,𝑥). The Lebesgue dominated convergence theorem for the Pettis integral then implies that for each 𝜑𝐸, 𝜑𝑦𝑛(𝑡)=𝜑𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑛(𝑠)𝑑𝑠𝜑𝑔(𝑡)+𝑇0;𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠(3.20) that is, 𝑦𝑛(𝑡)𝑁𝑥(𝑡) in (𝐸,𝑤). Repeating this for each 𝑡𝐽 shows 𝑦(𝑡)𝑁𝑥(𝑡).
Step  4. The implication (2.9) holds. Now let 𝑉 be a subset of 𝐷 such that 𝑉conv(𝑁(𝑉){0}). Clearly, 𝑉(𝑡)conv(𝑁(𝑉){0}) for all 𝑡𝐽. Hence, 𝑁𝑉(𝑡)𝑁𝐷(𝑡),𝑡𝐽, is bounded in 𝑃(𝐸).
Since function 𝑔 is continuous on 𝐽, the set {𝑔(𝑡),𝑡𝐽}𝐸 is compact, so 𝛽(𝑔(𝑡))=0. By assumption (H4) and the properties of the measure 𝛽, we have for each 𝑡𝐽𝛽(𝑁(𝑉)(𝑡))=𝛽𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠𝑣𝑆𝐹,𝑥,𝑥𝑉,𝑡𝐽𝛽{𝑔(𝑡)𝑡𝐽}+𝛽𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠𝑣𝑆𝐹,𝑥,𝑥𝑉,𝑡𝐽𝛽𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠𝑣(𝑡)𝐹(𝑡,𝑥(𝑡)),𝑥𝑉,𝑡𝐽𝑇0𝐺(𝑡,𝑠)𝑝𝑓𝑝(𝑠)𝛽(𝑉(𝑠))𝑑𝑠𝑓𝐿𝑇0𝑝𝐺(𝑡,𝑠)𝛽(𝑉(𝑠))𝑑𝑠𝑓𝐿𝐺𝑇0𝛽(𝑉(𝑠))𝑑𝑠,(3.21) which gives 𝑣𝑝𝑓𝐿𝑣𝐺.(3.22)
This means that 𝑣𝑝1𝑓𝐿𝐺0.(3.23) By (3.5) it follows that 𝑣=0; that is, 𝑣=0 for each 𝑡𝐽, 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 𝑐𝐷𝛼𝑢𝑛1(𝑡)7𝑒𝑡+13||𝑢1+𝑛||[](𝑡),𝑡𝐽=0,𝑇,1<𝛼2,𝑢(0)=𝜆1𝑢(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=𝑢𝑢=1,𝑢2,,𝑢𝑛,𝑛=1||𝑢𝑛||<(4.2) with the norm 𝑢𝐸=𝑛=1||𝑢𝑛||.(4.3) Set 𝑢𝑢=1,𝑢2,,𝑢𝑛𝑓,,𝑓=1,𝑓2,,𝑓𝑛,𝑓,𝑛𝑡,𝑢𝑛=17𝑒𝑡+13||𝑢1+𝑛||,𝑡𝐽.(4.4) For each 𝑢𝑛 and 𝑡𝐽, we have ||𝑓𝑛𝑡,𝑢𝑛||17𝑒𝑡+13||𝑢1+𝑛||.(4.5)
Hence conditions (H1), (H2), and (H3) hold with 𝑝𝑓(𝑡)=1/(7𝑒𝑡+13),𝑡𝐽, and 𝜓(𝑢)=1+𝑢,𝑢[0,). For any bounded set 𝐷𝑙1, we have 1𝛽(𝐹(𝑡,𝐷))7𝑒𝑡+13𝛽(𝐷),𝑡𝐽.(4.6) Hence (H4) is satisfied. From (3.8), we have 𝐺(𝑡,𝑠)=(𝑡𝑠)𝛼1Γ(𝛼)(1𝑠)𝛼1+2Γ(𝛼)(12𝑡)(1𝑠)𝛼2,4Γ(𝛼1)if0𝑠𝑡1,(1𝑠)𝛼1+2Γ(𝛼)(12𝑡)(1𝑠)𝛼2,4Γ(𝛼1)if0𝑡𝑠1.(4.7) So, we get 10𝐺(𝑡,𝑠)𝑑𝑠=𝑡0𝐺(𝑡,𝑠)𝑑𝑠+1𝑡=𝐺(𝑡,𝑠)𝑑𝑠𝑡0(𝑡𝑠)𝛼1Γ(𝛼)(1𝑠)𝛼1+2Γ(𝛼)(12𝑡)(1𝑠)𝛼2+4Γ(𝛼1)𝑑𝑠1𝑡(1𝑠)𝛼1+2Γ(𝛼)(12𝑡)(1𝑠)𝛼2=4Γ(𝛼1)𝑑𝑠4𝑡𝛼2+4Γ(𝛼+1)12𝑡.4Γ(𝛼)(4.8) A simple computation gives 𝐺<1+14Γ(𝛼)2Γ(𝛼+1)=𝐴𝛼.(4.9) We shall check that condition (3.5) is satisfied. Indeed 𝑝𝐿𝐺<17𝑒13𝐴𝛼<1,(4.10) which is satisfied for some 𝛼(1,2], and (H5) is satisfied for 𝑅>𝐴𝛼/(7𝑒13𝐴𝛼). 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.

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