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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Volume 2012 |Article ID 530624 | https://doi.org/10.1155/2012/530624

Wen-Xue Zhou, Hai-Zhong Liu, "Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions", Journal of Applied Mathematics, vol. 2012, Article ID 530624, 13 pages, 2012. https://doi.org/10.1155/2012/530624

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

Academic Editor: Ram N. Mohapatra
Received24 Apr 2012
Revised21 Jul 2012
Accepted22 Jul 2012
Published23 Aug 2012

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)),𝑡∈𝐽∶=0,𝑇,𝑇>0,𝑢(0)=𝜆1𝑢(𝑇)+𝜇1,𝑢′(0)=𝜆2𝑢′(𝑇)+𝜇2,𝜆1≠1,𝜆2≠1,(1.1) where 1<𝛼≤2 is a real number, 𝑐𝐷𝛼 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,ğœŽ(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 â€–ğ‘¦â€–âˆž=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 ğ¿âˆž(𝐽,𝐸) to be the Banach space of bounded measurable functions 𝑦∶𝐽→𝐸 equipped with the norm â€–ğ‘¦â€–ğ¿âˆž=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, ğ‘¦î…ž, is absolutely continuous.

Let (𝐸,‖⋅‖) be a Banach space, and let 𝑃cl(𝐸)={𝑌∈𝒫(𝐸)∶𝑌isclosed}, 𝑃𝑏(𝐸)={𝑌∈𝒫(𝐸)∶𝑌isbounded}, 𝑃cp(𝐸)={𝑌∈𝒫(𝐸)∶𝑌iscompact}, and 𝑃cp,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∈𝑃𝑏(𝐸) (i.e., sup𝑥∈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∈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., 𝑥𝑛→𝑥∗,𝑦𝑛→𝑦∗,𝑦𝑛∈𝐺(𝑥𝑛) 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→𝑃cl(𝐸) is said to be measurable if for every 𝑦∈𝐸, the function ||||𝑡⟼𝑑(𝑦,𝐺(𝑡))=inf𝑦−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 𝑆𝐹,𝑦 be the set of selections of 𝐹 defined by 𝑆𝐹,𝑦=𝑓∈𝐿1(𝐽,𝐸)∶𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡))a.e..𝑡∈𝐽(2.5)

Definition 2.1. A function â„Žâˆ¶ğ¸â†’ğ¸ 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 (𝑥𝑛)𝑛 in 𝐸 with 𝑥𝑛(𝑡)→𝑥(𝑡) in (𝐸,𝜔) then ℎ(𝑥𝑛(𝑡))→ℎ(𝑥(𝑡)) in (𝐸,𝜔) for each 𝑡→𝐽).

Definition 2.2. A function 𝐹∶𝑄→𝑃cl,cv(𝑄) has a weakly sequentially closed graph if for any sequence (𝑥𝑛,𝑦𝑛)∞1∈𝑄×𝑄, 𝑦𝑛∈𝐹(𝑥𝑛) for 𝑛∈{1,2,…} with 𝑥𝑛(𝑡)→𝑥(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽 and 𝑦𝑛(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 𝑥𝐽∈𝐸 corresponding to each 𝐼⊂𝐽 such that 𝜑(𝑥𝐼∫)=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, 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 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, 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 𝛽∶Ω𝐸→[0,∞) defined by 𝛽(𝑋)=inf𝜖>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(𝑆𝜔)=𝛽(𝑆), where 𝑆𝜔 denotes the weak closure of 𝑆;(e)𝛽(𝑆+𝑇)≤𝛽(𝑆)+𝛽(𝑇);(f)𝛽(ğ‘Žğ‘†)=|ğ‘Ž|𝛼(𝑆);(g)𝛽(conv(𝑆))=𝛽(𝑆);(h)𝛽(∪|𝜆|â‰¤â„Žğœ†ğ‘†)=â„Žğ›½(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 â„Žâˆ¶ğ½â†’ğ¸ be a function. The fractional Pettis integral of the function ℎ of order 𝛼∈ℝ+ is defined by ğ¼ğ›¼î€œâ„Ž(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 â„Žâˆ¶ğ½â†’ğ¸, the Caputo fractional-order derivative of ℎ is defined by î€·ğ‘ğ·ğ›¼ğ‘Ž+ℎ(1𝑡)=Γ(𝑛−𝛼)ğ‘¡ğ‘Ž(𝑡−𝑠)𝑛−𝛼−1ℎ(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→𝑃cl,cv(𝑄) has a weakly sequentially closed graph. If the implication 𝑉=conv({0}∪𝐹(𝑉))âŸ¹ğ‘‰ğ‘–ğ‘ ğ‘Ÿğ‘’ğ‘™ğ‘Žğ‘¡ğ‘–ğ‘£ğ‘’ğ‘™ğ‘¦ğ‘¤ğ‘’ğ‘Žğ‘˜ğ‘™ğ‘¦ğ‘ğ‘œğ‘šğ‘ğ‘Žğ‘ğ‘¡(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 𝑐𝐷𝛼𝑦(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→𝑃cp,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 ğ‘ğ‘“âˆˆğ¿âˆž(𝐽,ℝ+) and a continuous nondecreasing function 𝜓∶[0,∞)→[0,∞) such that ‖𝐹(𝑡,𝑢)‖=sup{|𝑣|∶𝑣∈𝐹(𝑡,𝑢)}≤𝑝𝑓(𝑡)𝜓(‖𝑢‖);(3.2)(H4)for each bounded set 𝐷⊂𝐸, and each 𝑡∈𝐼, the following inequality holds: 𝛽(𝐹(𝑡,𝐷))≤𝑝𝑓(𝑡)⋅𝛽(𝐷);(3.3)(H5)there exists a constant 𝑅>0 such that 𝑅𝑔∗+â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆžğœ“(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 â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆžğºâˆ—<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] 𝑐𝐷𝛼𝑢(𝑡)=𝜌(𝑡),𝑡∈(0,𝑇),1<𝛼≤2𝑢(𝑡)=𝜆1𝑢(𝑇)+𝜇1,ğ‘¢î…ž(0)=𝜆2ğ‘¢î…ž(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−1−𝜆Γ(𝛼)1(𝑇−𝑠)𝛼−1𝜆1+𝜆−1Γ(𝛼)2𝜆1𝑇+1−𝜆1𝑡(𝑇−𝑠)𝛼−2𝜆2𝜆−11,−1Γ(𝛼−1)if−𝜆0≤𝑠≤𝑡≤𝑇,1(𝑇−𝑠)𝛼−1𝜆1+𝜆−1Γ(𝛼)2𝜆1𝑇+1−𝜆1𝑡(𝑇−𝑠)𝛼−2𝜆2𝜆−11,−1Γ(𝛼−1)if𝜇0≤𝑡≤𝑠≤𝑇,𝑔(𝑡)=2𝜆1𝑇+1−𝜆1𝑡𝜆2𝜆−11−𝜇−11𝜆1.−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,𝑡∈𝐽,𝑔∗=max0≤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)→𝑃cl,cv(𝐶(𝐽,𝐸)) defined by 𝑁(𝑥)=â„Žâˆˆğ¶(𝐽,𝐸)∶ℎ(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠,𝑣∈𝑆𝐹,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,⋅)âˆˆğ¿âˆž(𝐽), then 𝐺(𝑡,⋅)𝑣(⋅) is Pettis integrable and thus 𝑁 is well defined.
Let 𝑅>0, and consider the set 𝐷=𝑥∈𝐶(𝐽,𝐸)âˆ¶â€–ğ‘¥â€–âˆžâ€–â€–ğ‘¥î€·ğ‘¡â‰¤ğ‘…,1𝑡−𝑥2‖‖≤‖‖𝑔𝑡1𝑡−𝑔2‖‖+â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆžî€œğœ“(𝑅)𝑇0‖‖𝐺𝑡2𝑡,𝑠−𝐺1‖‖,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 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 â„Žğ‘–î€œ(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑖(𝑠)𝑑𝑠,𝑖=1,2.(3.12) Let 0≤𝑑≤1. Then, for each 𝑡∈𝐽, we have î€ºğ‘‘â„Ž1+(1−𝑑)ℎ2(𝑡)=𝑔(𝑡)+𝑇0𝐺(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 ğ‘‘â„Ž1+(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 𝜑𝑠∈𝐸∗ with ‖𝜑𝑠‖=1 and 𝜑𝑠(𝑢(𝑠))=‖𝑢(𝑠)‖. Hence, for each fixed 𝑡∈𝐽, we have ‖𝑢(𝑡)‖=𝜑𝑡(𝑢(𝑡))=𝜑𝑡𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠≤𝜑𝑡(𝑔(𝑡))+𝜑𝑡𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠≤‖𝑔(𝑡)‖+𝑇0‖𝐺(𝑡,𝑠)‖𝜑𝑡(𝑣(𝑠))𝑑𝑠≤𝑔∗+ğºâˆ—ğœ“î€·â€–ğ‘¥â€–âˆžî€¸â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆž.(3.14)
Therefore, by (H5), we have â€–ğ‘¢â€–âˆžâ‰¤ğ‘”âˆ—+â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆžğºâˆ—ğœ“î€·â€–ğ‘…â€–âˆžî€¸â‰¤ğ‘….(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𝜏−𝑢1‖‖𝑔𝑡=𝜑2𝑡−𝑔1+𝑇0𝐺𝜏2𝜏,𝑠−𝐺1𝑔𝑡,𝑠⋅𝑣(𝑠)𝑑𝑠≤𝜑2𝑡−𝑔1+𝜑𝑇0𝐺𝜏2𝜏,𝑠−𝐺1≤‖‖𝑔𝑡,𝑠⋅𝑣(𝑠)𝑑𝑠2𝑡−𝑔1‖‖+𝑇0‖‖𝐺𝜏2𝜏,𝑠−𝐺1‖‖‖≤‖‖𝑔𝑡,𝑠⋅‖𝑣(𝑠)𝑑𝑠2𝑡−𝑔1‖‖‖‖𝑝+𝜓(𝑅)ğ‘“â€–â€–ğ¿âˆžî€œğ‘‡0‖‖𝐺𝜏2𝜏,𝑠−𝐺1‖‖,𝑠𝑑𝑠;(3.16) this means that 𝑢∈𝐷.
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 𝑡∈𝐽, 𝑦𝑛(𝑡)→𝑦(𝑡) in (𝐸,𝜔) for each 𝑡∈𝐽, and 𝑦𝑛∈𝑁(𝑥𝑛) for 𝑛∈{1,2,…}. We will show that 𝑦∈𝑁𝑥. By the relation 𝑦𝑛∈𝑁(𝑥𝑛), we mean that there exists 𝑣𝑛∈𝑆𝐹,𝑥𝑛 such that 𝑦𝑛(𝑡)=𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑛(𝑠)𝑑𝑠.(3.17)
We must show that there exists 𝑣∈𝑆𝐹,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 𝑣𝑛𝑚 such that 𝑣𝑛𝑚(⋅)⟶𝑣(⋅)in(𝐸,𝜔)asğ‘£ğ‘šâŸ¶âˆžğ‘›ğ‘šî€·(𝑡)∈𝐹𝑡,𝑥𝑛(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𝑦𝑛(𝑡)=𝜑𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣𝑛(𝑠)𝑑𝑠⟶𝜑𝑔(𝑡)+𝑇0;𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠(3.20) that is, 𝑦𝑛(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∈𝐽𝛽(𝑁(𝑉)(𝑡))=𝛽𝑔(𝑡)+𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠∶𝑣∈𝑆𝐹,𝑥,𝑥∈𝑉,𝑡∈𝐽≤𝛽{𝑔(𝑡)∶𝑡∈𝐽}+𝛽𝑇0𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠∶𝑣∈𝑆𝐹,𝑥,𝑥∈𝑉,𝑡∈𝐽≤𝛽𝑇0≤𝐺(𝑡,𝑠)𝑣(𝑠)𝑑𝑠∶𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)),𝑥∈𝑉,𝑡∈𝐽𝑇0‖𝐺(𝑡,𝑠)‖⋅𝑝𝑓≤‖‖𝑝(𝑠)⋅𝛽(𝑉(𝑠))ğ‘‘ğ‘ ğ‘“â€–â€–ğ¿âˆžâ‹…î€œğ‘‡0≤‖‖𝑝‖𝐺(𝑡,𝑠)‖⋅𝛽(𝑉(𝑠))ğ‘‘ğ‘ ğ‘“â€–â€–ğ¿âˆžâ‹…ğºâˆ—â‹…î€œğ‘‡0𝛽(𝑉(𝑠))𝑑𝑠,(3.21) which gives â€–ğ‘£â€–âˆžâ‰¤â€–â€–ğ‘ğ‘“â€–â€–ğ¿âˆžâ‹…â€–ğ‘£â€–âˆžâ‹…ğºâˆ—.(3.22)
This means that â€–ğ‘£â€–âˆžâ‹…î€ºâ€–â€–ğ‘1âˆ’ğ‘“â€–â€–ğ¿âˆžâ‹…ğºâˆ—î€»â‰¤0.(3.23) By (3.5) it follows that â€–ğ‘£â€–âˆž=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 𝑐𝐷𝛼𝑢𝑛1(𝑡)∈7𝑒𝑡+13||𝑢1+𝑛||[](𝑡),𝑡∈𝐽∶=0,𝑇,1<𝛼≤2,𝑢(0)=𝜆1𝑢(𝑇)+𝜇1,ğ‘¢î…ž(0)=𝜆2ğ‘¢î…ž(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∶,â€¦âˆžî“ğ‘›=1||𝑢𝑛||<∞(4.2) with the norm ‖𝑢‖𝐸=âˆžî“ğ‘›=1||𝑢𝑛||.(4.3) Set 𝑢𝑢=1,𝑢2,…,𝑢𝑛𝑓,…,𝑓=1,𝑓2,…,𝑓𝑛,𝑓,…𝑛𝑡,𝑢𝑛=17𝑒𝑡+13||𝑢1+𝑛||,𝑡∈𝐽.(4.4) For each 𝑢𝑛∈ℝ and 𝑡∈𝐽, we have ||𝑓𝑛𝑡,𝑢𝑛||≤17𝑒𝑡+13||𝑢1+𝑛||.(4.5)
Hence conditions (H1), (H2), and (H3) hold with 𝑝𝑓(𝑡)=1/(7𝑒𝑡+13),𝑡∈𝐽, and 𝜓(𝑢)=1+𝑢,𝑢∈[0,∞). For any bounded set 𝐷⊂𝑙1, we have 1𝛽(𝐹(𝑡,𝐷))≤7𝑒𝑡+13⋅𝛽(𝐷),∀𝑡∈𝐽.(4.6) Hence (H4) is satisfied. From (3.8), we have ⎧⎪⎨⎪⎩𝐺(𝑡,𝑠)=(𝑡−𝑠)𝛼−1Γ−(𝛼)(1−𝑠)𝛼−1+2Γ(𝛼)(1−2𝑡)(1−𝑠)𝛼−2,4Γ(𝛼−1)if−0≤𝑠≤𝑡≤1,(1−𝑠)𝛼−1+2Γ(𝛼)(1−2𝑡)(1−𝑠)𝛼−2,4Γ(𝛼−1)if0≤𝑡≤𝑠≤1.(4.7) So, we get 10𝐺(𝑡,𝑠)𝑑𝑠=𝑡0𝐺(𝑡,𝑠)𝑑𝑠+1𝑡=𝐺(𝑡,𝑠)𝑑𝑠𝑡0(𝑡−𝑠)𝛼−1Γ−(𝛼)(1−𝑠)𝛼−1+2Γ(𝛼)(1−2𝑡)(1−𝑠)𝛼−2+4Γ(𝛼−1)𝑑𝑠1𝑡−(1−𝑠)𝛼−1+2Γ(𝛼)(1−2𝑡)(1−𝑠)𝛼−2=4Γ(𝛼−1)𝑑𝑠4𝑡𝛼−2+4Γ(𝛼+1)1−2𝑡.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 â€–ğ‘â€–ğ¿âˆžğºâˆ—<17𝑒13𝐴𝛼<1,(4.10) which is satisfied for some 𝛼∈(1,2], and (H5) is satisfied for 𝑅>𝐴𝛼/(7𝑒13−𝐴𝛼). 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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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.


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