JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 530624 10.1155/2012/530624 530624 Research Article Existence of Weak Solutions for Nonlinear Fractional Differential Inclusion with Nonseparated Boundary Conditions Zhou Wen-Xue 1, 2 Liu Hai-Zhong 1 Mohapatra Ram N. 1 Department of Mathematics Lanzhou Jiaotong University Lanzhou 730070 China lzjtu.edu.cn 2 College of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049 China xjtu.edu.cn 2012 23 8 2012 2012 24 04 2012 21 07 2012 22 07 2012 2012 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.

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: (1.1)Dαcu(t)F(t,u(t)),tJ:=[0,T],T>0,u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2,λ11,λ21, where 1<α2 is a real number,   cDα is the Caputo fractional derivative. F:J×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  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  and subsequently developed and used in many papers; see, for example, Banaś and Sadarangani , Guo et al. , Krzyśka and Kubiaczyk , Lakshmikantham and Leela , Mönch’s , O’Regan [25, 26], Szufla [27, 28], and the references therein.

In 2007, Ouahab  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  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 ; 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  and . About other results on fractional differential inclusions, we refer the reader to . 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 (2.1)y=sup{y(t):0tT}, and let L1(J,E) denote the Banach space of functions y:JE that are the Lebesgue integrable with norm (2.2)yL1=0Ty(t)dt. We let L(J,E) to be the Banach space of bounded measurable functions y:JE equipped with the norm (2.3)yL=inf{c>0:y(t)c,a.e.tJ}. Also, AC1(J,E) will denote the space of functions y:JE 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 G:EP(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{y:yG(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 y*G(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 G:JPcl(E) is said to be measurable if for every yE, the function (2.4)td(y,G(t))=inf{|y-z|:zG(t)} is measurable. For more details on multivalued maps, see the books of Aubin and Cellina , Aubin and Frankowska , Deimling , Hu and Papageorgiou , Kisielewicz , and Covitz and Nadler .

Moreover, for a given set V of functions v:J, 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 (2.5)SF,y={fL1(J,E):f(t)F(t,y(t))a.e.tJ}.

Definition 2.1.

A function h:EE 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 tJ).

Definition 2.2.

A function F:QPcl,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 [<xref ref-type="bibr" rid="B41">41</xref>]).

The function x:JE 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 [<xref ref-type="bibr" rid="B41">41</xref>]).

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 [<xref ref-type="bibr" rid="B42">42</xref>]).

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 (2.6)β(X)=inf{ϵ>0:thereexistsaweaklycompactsubsetΩofEsuchthatXϵB1+Ω}.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B42">42</xref>]).

The De Blasi measure of noncompactness satisfies the following properties:

STβ(S)β(T);

β(S)=0S is relatively weakly compact;

β(ST)=max{β(S),β(T)};

β(S¯ω)=β(S), where S¯ω denotes the weak closure of S;

β(S+T)β(S)+β(T);

β(aS)=|a|α(S);

β(conv(S))=β(S);

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 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 [<xref ref-type="bibr" rid="B25">25</xref>]).

Let h:JE be a function. The fractional Pettis integral of the function h of order α+ is defined by (2.7)Iαh(t)=0t(t-s)α-1Γ(α)h(s)ds, where the sign “” denotes the Pettis integral and Γ is the gamma function.

Definition 2.9 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

For a function h:JE, the Caputo fractional-order derivative of h is defined by (2.8)(Da+αch)(t)=1Γ(n-α)at(t-s)n-α-1h(n)(s)ds,n-1<α<n, where n=[α]+1 and [α] denotes the integer part of α.

Lemma 2.10 (see [<xref ref-type="bibr" rid="B43">43</xref>]).

Let E be a Banach space with Q a nonempty, bounded, closed, convex, equicontinuous subset of C(J,E). Suppose F:QPcl,cv(Q) has a weakly sequentially closed graph. If the implication (2.9)V¯=conv¯({0}F(V))Visrelativelyweaklycompact 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 (3.1)  cDαy(t)=v(t)a.e.tJ,1<α2, 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:

F:J×EPcp,cv(E) has weakly sequentially closed graph;

for each continuous xC(J,E), there exists a scalarly measurable function v:JE with v(t)F(t,x(t)) a.e. on J and v is Pettis integrable on J;

there exist pfL(J,+) and a continuous nondecreasing function ψ:[0,)[0,) such that (3.2)F(t,u)=sup{|v|:vF(t,u)}pf(t)ψ(u);

for each bounded set DE, and each tI, the following inequality holds: (3.3)β(F(t,D))pf(t)β(D);

there exists a constant R>0 such that (3.4)Rg*+pfLψ(R)G*>1,

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 (3.5)pfLG*<1, 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  (3.6)Dαcu(t)=ρ(t),t(0,T),1<α2u(t)=λ1u(T)+μ1,  u(0)=λ2u(T)+μ2,λ11,λ21 has a unique solution (3.7)u(t)=0TG(t,s)ρ(s)ds+g(t), where G(t,s) is defined by the formula (3.8)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),if0stT,-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),if0tsT,g(t)=μ2[λ1T+(1-λ1)t](λ2-1)(λ1-1)-μ1λ1-1.

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 (3.9)G*=sup{0T|G(t,s)|ds,tJ},g*=max0tTg(t).

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 (3.10)N(x)={hC(J,E):h(t)=g(t)+0TG(t,s)v(s)ds,vSF,x}, and refer to  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 v:JE 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 (3.11)D={0TxC(J,E):xR,x(t1)-x(t2)g(t1)-g(t2)+pfLψ(R)0TG(t2,s)-G(t1,s)dsfort1,t2J}; 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 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 tJ, we have (3.12)hi(t)=g(t)+0TG(t,s)vi(s)ds,i=1,2. Let 0d1. Then, for each tJ, we have (3.13)[dh1+(1-d)h2](t)=g(t)+0TG(t,s)[dv1(s)+(1-d)v2(s)]ds. Since F has convex values, (dv1+(1-d)v2)(t)F(t,y) and we have dh1+(1-d)h2N(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 v:JE 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 (3.14)u(t)=φt(u(t))=φt(g(t)+0TG(t,s)v(s)ds)φt(g(t))+φt(0TG(t,s)v(s)ds)g(t)+0TG(t,s)φt(v(s))dsg*+G*ψ(x)pfL.

Therefore, by (H5), we have (3.15)ug*+pfLG*ψ(R)R.

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, (3.16)u(τ2)-u(τ1)=φ(g(t2)-g(t1)+0T[G(τ2,s)-G(τ1,s)]v(s)ds)φ(g(t2)-g(t1))+φ(0T[G(τ2,s)-G(τ1,s)]v(s)ds)g(t2)-g(t1)+0TG(τ2,s)-G(τ1,s)v(s)dsg(t2)-g(t1)+ψ(R)pfL0TG(τ2,s)-G(τ1,s)ds; 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 (3.17)yn(t)=g(t)+0TG(t,s)vn(s)ds.

We must show that there exists vSF,x such that, for each tJ, (3.18)y(t)=g(t)+0TG(t,s)v(s)ds.

Since F(·,·) has compact values, there exists a subsequence vnm such that (3.19)vnm()v()in(E,ω)asmvnm(t)F(t,xn(t))a.e.tJ. 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*, (3.20)φ(yn(t))=φ(g(t)+0TG(t,s)vn(s)ds)φ(g(t)+0TG(t,s)v(s)ds); 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 tJ(3.21)β(N(V)(t))=β{g(t)+0TG(t,s)v(s)ds:vSF,x,xV,tJ}β{g(t):tJ}+β{0TG(t,s)v(s)ds:vSF,x,xV,tJ}β{0TG(t,s)v(s)ds:v(t)F(t,x(t)),xV,tJ}0TG(t,s)pf(s)β(V(s))dspfL0TG(t,s)β(V(s))dspfLG*0Tβ(V(s))ds, which gives (3.22)vpfLvG*.

This means that (3.23)v[1-pfLG*]0. 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 (4.1)(Dαcun)(t)17et+13(1+|un(t)|),tJ:=[0,T],1<α2,u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2,

Set T=1, λ1=λ2=-1, μ1=μ2=0, then g(t)=0. So g*=0.

Let (4.2)E=l1={u=(u1,u2,,un,):n=1|un|<} with the norm (4.3)uE=n=1|un|. Set (4.4)u=(u1,u2,,un,),f=(f1,f2,,fn,),fn(t,un)=17et+13(1+|un|),tJ. For each un and tJ, we have (4.5)|fn(t,un)|17et+13(1+|un|).

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 (4.6)β(F(t,D))17et+13β(D),tJ. Hence (H4) is satisfied. From (3.8), we have (4.7)G(t,s)={(t-s)α-1Γ(α)-(1-s)α-12Γ(α)+(1-2t)(1-s)α-24Γ(α-1),if0st1,-(1-s)α-12Γ(α)+(1-2t)(1-s)α-24Γ(α-1),if0ts1. So, we get (4.8)01G(t,s)ds=0tG(t,s)ds+t1G(t,s)ds=0t[(t-s)α-1Γ(α)-(1-s)α-12Γ(α)+(1-2t)(1-s)α-24Γ(α-1)]ds+t1[-(1-s)α-12Γ(α)+(1-2t)(1-s)α-24Γ(α-1)]ds=4tα-24Γ(α+1)+1-2t4Γ(α). A simple computation gives (4.9)G*<14Γ(α)+12Γ(α+1)Aα. We shall check that condition (3.5) is satisfied. Indeed (4.10)pLG*<17e13Aα<1, 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.