DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 527969 10.1155/2012/527969 527969 Research Article Weak Solutions for Nonlinear Fractional Differential Equations in Banach Spaces Zhou Wen-Xue 1, 2 Chang Ying-Xiang 1 Liu Hai-Zhong 1 Sivasundaram Seenith 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 16 7 2012 2012 26 02 2012 12 04 2012 06 05 2012 2012 Copyright © 2012 Wen-Xue Zhou et al. 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 weak solutions for a nonlinear boundary value problem of fractional differential equations in Banach space. Our analysis relies on the Mönch's 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 equation: (1.1)cD0+αu(t)=f(t,u(t)),tJ:=[0,T],u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2,λ11,λ21, where 1<α2 is a real number, cD0+α is the Caputo's fractional derivative, λ1,λ2,μ1,μ2. f:J×EE is a given function satisfying some assumptions that will be specified later, and E is a Banach space with norm u.

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 , O'Regan [26, 27], Szufla [28, 29], and the references therein. As far as we know, there are very few results devoted to weak solutions of nonlinear fractional differential equations . 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's fixed point theorem combined with the technique of measures of weak noncompactness. Our results can be seen as a supplement of the results in  (see Remark 3.8).

The remainder of this is organized as follows. In Section 2, we provide some basic definitions, preliminaries facts, and various lemmas which are needed later. In Section 3, we give main results of problem (1.1). In the end, we also give an example for the illustration of the theories established in this paper.

2. Preliminaries and Lemmas

In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.

Let J:=[0,T] and L1(J,E) denote the Banach space of real-valued Lebesgue integrable functions on the interval J, L(J,E) denote the Banach space of real-valued essentially bounded and measurable functions defined over J with the norm ·L.

Let E be a real reflexive Banach space with norm · and dual E*, and let (E,ω)=(E,σ(E,E*)) denote the space E with its weak topology. Here, C(J,E) is the Banach space of continuous functions x:JE with the usual supremum norm x:=sup{x(t):tJ}.

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}.

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 (see [<xref ref-type="bibr" rid="B33">33</xref>]).

The function x:JE is said to be 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 2.3 (see [<xref ref-type="bibr" rid="B33">33</xref>]).

If x(·) is Pettis integrable and h(·) is a measurable and essentially bounded real-valued function, then x(·)h(·) is Pettis integrable.

Definition 2.4 (see [<xref ref-type="bibr" rid="B34">34</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 β(X)=inf{ϵ>0: there exists a weakly compact subset Ω of E such that XϵB1+Ω}.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

The De sBlasi 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 2.6.

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 2.7 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

Let h:JE be a function. The fractional Pettis integral of the function h of order α+ is defined by (2.1)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 2.8 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

For a function f:JE, the Caputo fractional-order derivative of f is defined by (2.2)(Dca+αf)(t)=1Γ(n-α)at(t-s)n-α-1f(n)(s)ds,n-1<α<n, where n=[α]+1 and [α] denotes the integer part of α.

Lemma 2.9 (see [<xref ref-type="bibr" rid="B28">28</xref>]).

Let D be a closed convex and equicontinuous subset of a metrizable locally convex vector space C(J,E) such that 0D. Assume that A:DD is weakly sequentially continuous. If the implication (2.3)V-= conv ¯({0}A(V))Visrelativelyweaklycompact, holds for every subset V of D, then A has a fixed point.

3. Main Results

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

Definition 3.1.

A function xC(J,Eω) is said to be a solution of the problem (1.1) if x satisfies the equation cD0+αu(t)=f(t,u(t)) on J and satisfies the conditions u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2.

For the existence results on the problem (1.1), we need the following auxiliary lemmas.

Lemma 3.2 (see [<xref ref-type="bibr" rid="B3">3</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

For α>0, the general solution of the fractional differential equation cD0+αu(t)=0 is given by (3.1)h(t)=C0+C1t+C2t2++Cn-1tn-1,CiR,i=0,1,2,,n-1,n=[α]+1.

Lemma 3.3 (see [<xref ref-type="bibr" rid="B3">3</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

Assume that hC(0,1)L(0,1) with a fractional derivative of order α>0 that belongs to C(0,1)L(0,1). Then (3.2)I0+αcD0+αh(t)=h(t)+c0+c1t+c2t2++cn-1tn-1 for some ci, i=0,1,2,,n-1, where n=[α]+1.

We derive the corresponding Green's function for boundary value problem (1.1) which will play major role in our next analysis.

Lemma 3.4.

Let ρC(J,E) be a given function, then the boundary-value problem (3.3)cD0+αu(t)=ρ(t),t(0,T),1<α2,u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2,λ11,λ21 has a unique solution (3.4)u(t)=0TG(t,s)ρ(s)ds+μ2[λ1T+(1-λ1)t](λ1-1)(λ2-1)-μ1(λ1-1), where G(t,s) is defined by the formula (3.5)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(λ1-1)(λ2-1)Γ(α-1), if   0stT,-λ1(T-s)α-1(λ1-1)Γ(α)+λ2[λ1T+(1-λ1)t](T-s)α-2(λ1-1)(λ2-1)Γ(α-1), if   0tsT. Here G(t,s) is called the Green's function of boundary value problem (3.3).

Proof.

By the Lemma 3.3, we can reduce the equation of problem (3.3) to an equivalent integral equation (3.6)u(t)=I0+αρ(t)-c1-c2t=1Γ(α)0t(t-s)α-1ρ(s)ds-c1-c2t for some constants c1,c2. On the other hand, by relations D0+αI0+αu(t)=u(t) and I0+mI0+nu(t)=I0+m+nu(t), for m,n>0,uL(0,1), we have (3.7)u(t)=-c2+1Γ(α-1)0t(t-s)α-2ρ(s)ds=-c2+I0+α-1ρ(t).

Applying the boundary conditions (3.3), we have (3.8)c1=λ1λ1-1[0T(T-s)α-1Γ(α)ρ(s)ds-Tλ2λ2-1(0T(T-s)α-2Γ(α-1)ρ(s)ds+μ2λ2)+μ1λ1],    c2=λ2(λ2-1)1Γ(α-1)0T(T-s)α-2ρ(s)ds+μ2(λ2-1).

Therefore, the unique solution of problem (3.3) is (3.9)u(t)=1Γ(α)0t(t-s)α-1ρ(s)ds-c1-c2t=1Γ(α)0t(t-s)α-1ρ(s)ds-λ1λ1-1[0T(T-s)α-1Γ(α)ρ(s)ds  -Tλ2λ2-1(0T(T-s)α-2Γ(α-1)ρ(s)ds+μ2λ2)+μ1λ1]-[λ2(λ2-1)1Γ(α-1)0T(T-s)α-2ρ(s)ds+μ2(λ2-1)]t=0TG(t,s)ρ(s)ds+μ2[λ1T+(1-λ1)t](λ1-1)(λ2-1)  -μ1(λ1-1), which completes the proof.

Remark 3.5.

From the expression of G(t,s), it is obvious that G(t,s) is continuous on J×J. Denote (3.10)G*=sup{0T|G(t,s)|ds:tJ}.

Remark 3.6.

Letting ξ1=1/(λ1-1), ξ2=1/(λ1-1)(λ2-1), g(t)=μ2[λ1T+(1-λ1)t]/(λ1-1)(λ2-1)-μ1/(λ1-1)=μ2[λ1T+(1-λ1)t]ξ2-μ1ξ1, it is obvious that g(t) is continuous in J, denoting g*=sup{|g(t)|,tJ}.

To prove the main results, we need the following assumptions:

for each tJ, the function f(t,·) is weakly sequentially continuous;

for each xC(J,E), the function f(·,x(·)) is Pettis integrable on J;

there exists pfL(J,+) such that f(t,u)pf(t)u, for a.e. tJ and each uE;

there exists pfL(J,E) and a continuous nondecreasing function ψ:[0,)(0,) such that f(t,u)pf(t)ψ(u), for a.e. tJ and each uE;

for each bounded set DE, and each tJ, the following inequality holds (3.11)β(f(t,D))pf(t)β(D);

there exists a constant R>0 such that (3.12)Rg*+pfLψ(R)G*>1,

where pfL=sup{pf(t):tJ}.

Theorem 3.7.

Let E be a reflexive Banach space and assume that (H1)–(H3) are satisfied. If (3.13)pfLG*<1, then the problem (1.1) has at least one solution on J.

Proof.

Let the operator 𝒜:C(J,E)C(J,E) defined by the formula (3.14)(Au)(t):=0TG(t,s)f(s,u(s))ds+g(t), where G(·,·) is the Green's function defined by (3.5). It is well known the fixed points of the operator 𝒜 are solutions of the problem (1.1).

First notice that, for xC(J,E), we have f(·,x(·))P(J,E) (assumption (H2)). Since, sG(t,s)L(J), then G(t,·)f(·,x(·)) is Pettis integrable for all tJ by Lemma 2.3, and so the operator 𝒜 is well defined.

Let (3.15)Rg*1-pfLG*, and consider the set (3.16)D={0T|G(t2,s)-G(t1,s)|dsxC(J,E):xR,x(t1)-x(t2)|μ2(1-λ1)ξ2||t2-t1|+RpfL0T|G(t2,s)-G(t1,s)|dsfor  t1,t2J}. Clearly, the subset D is closed, convex, and equicontinuous. We shall show that 𝒜 satisfies the assumptions of Lemma 2.9. The proof will be given in three steps.

Step  1. We will show that the operator 𝒜 maps D into itself.

Take xD, tJ and assume that 𝒜x(t)0. Then there exists ψE* such that 𝒜x(t)=ψ(𝒜x(t)). Thus (3.17)(Ax)(t)=ψ((Ax)(t))=ψ(g(t)+0TG(t,s)f(s,y(s))ds)ψ(g(t))+0T|G(t,s)|ψ(f(s,x(s)))dsg(t)+0T|G(t,s)|pf(s)x(s)dsg*+pfLRG*R.

Let τ1,τ2J,  τ1<τ2 and xD, so 𝒜x(τ2)-𝒜x(τ1)0. Then there exists ψE*, such that 𝒜x(τ2)-𝒜x(τ1)=ψ(𝒜x(τ2)-𝒜x(τ1)). Hence, (3.18)Ax(τ2)-Ax(τ1)=ψ(g(τ2)-g(τ1)+0T[G(τ2,s)-G(τ1,s)]f(s,x(s))ds)ψ(g(τ2)-g(τ1))+0T|G(τ2,s)-G(τ1,s)|f(s,x(s))dsg(τ2)-g(τ1)+RpfL0T|G(τ2,s)-G(τ1,s)|ds|μ2(1-λ1)ξ2||t2-t1|+RpfL0T|G(τ2,s)-G(τ1,s)|ds; this means that 𝒜(D)D.

Step  2. We will show that the operator 𝒜 is weakly sequentially continuous.

Let (xn) be a sequence in D and let (xn(t))x(t) in (E,w) for each tJ. Fix tJ. Since f satisfies assumptions (H1), we have f(t,xn(t)) converge weakly uniformly to f(t,x(t)). Hence, the Lebesgue Dominated Convergence Theorem for Pettis integrals implies that 𝒜xn(t) converges weakly uniformly to 𝒜x(t) in Eω. Repeating this for each tJ shows 𝒜xn𝒜x. Then 𝒜:DD is weakly sequentially continuous.

Step  3. The implication (2.3) holds. Now let V be a subset of D such that Vconv¯(𝒜(V){0}). Clearly, V(t)conv¯(𝒜(V){0}) for all tJ. Hence, 𝒜V(t)𝒜D(t),tJ, is bounded in E. Thus, 𝒜V(t) is weakly relatively compact since a subset of a reflexive Banach space is weakly relatively compact if and only if it is bounded in the norm topology. Therefore, (3.19)v(t)β(A(V)(t){0})β(A(V)(t))=0, thus, V is relatively weakly compact in E. In view of Lemma 2.9, we deduce that 𝒜 has a fixed point which is obviously a solution of the problem (1.1). This completes the proof.

Remark 3.8.

In the Theorem 3.7, we presented an existence result for weak solutions of the problem (1.1) in the case where the Banach space E is reflexive. However, in the nonreflexive case, conditions (H1)–(H3) are not sufficient for the application of Lemma 2.9; the difficulty is with condition (2.3). Our results can be seen as a supplement of the results in  (see Remark 3.8).

Theorem 3.9.

Let E be a Banach space, and assume assumptions (H1), (H2), (H3), (H4) are satisfied. If (3.13) holds, then the problem (1.1) has at least one solution on J.

Theorem 3.10.

Let E be a Banach space, and assume assumptions (H1), (H2), (H3)’, (H4), (H5) are satisfied. If (3.13) holds, then the problem (1.1) has at least one solution on J.

Proof.

Assume that the operator 𝒜:C(J,E)C(J,E) is defined by the formula (3.14). It is well known the fixed points of the operator 𝒜 are solutions of the problem (1.1).

First notice that, for xC(J,E), we have f(·,x(·))P(J,E) (assumption (H2)). Since, sG(t,s)L(J), then G(t,·)f(·,x(·)) for all tJ is Pettis integrable (Lemma 2.3) and thus the operator 𝒜 makes sense.

Let R>0, and consider the set (3.20)D={0T|G(t2,s)-G(t1,s)|dsfort1,t2JxC(J,E):xR,x(t1)-x(t2)|μ2(1-λ1)ξ2||t2-t1|+pfLψ(R)0T|G(t2,s)-G(t1,s)|dsfort1,t2J}. Clearly the subset 𝒟 is closed, convex and equicontinuous. We shall show that 𝒜 satisfies the assumptions of Lemma 2.9. The proof will be given in three steps.

Step  1. We will show that the operator 𝒜 maps 𝒟 into itself.

Take x𝒟, tJ and assume that 𝒜x(t)0. Then there exists ψE* such that 𝒜x(t)=ψ(𝒜x(t)). Thus (3.21)(Ax)(t)=ψ((Ax)(t))=ψ(g(t)+0TG(t,s)f(s,y(s))ds)ψ(g(t))+0T|G(t,s)|ψ(f(s,x(s)))dsψ(g(t))+0T|G(t,s)|pf(s)ψ(x(s))dsg*+pfLψ(R)G*R.

Let τ1,τ2J,τ1<τ2 and x𝒟, so 𝒜x(τ2)-𝒜x(τ1)0. Then there exist ψE* such that (3.22)Ax(τ2)-Ax(τ1)=ψ(Ax(τ2)-Ax(τ1)).

Thus (3.23)Ax(τ2)-Ax(τ1)=ψ(g(τ2)-g(τ1)+0T[G(τ2,s)-G(τ1,s)]f(s,x(s))ds)ψ(g(τ2)-g(τ1))+0T|G(τ2,s)-G(τ1,s)|  f(s,x(s))dsg(τ2)-g(τ1)+ψ(R)pfL0T|G(τ2,s)-G(τ1,s)|ds|μ2(1-λ1)ξ2||t2-t1|+ψ(R)pfL0T|G(τ2,s)-G(τ1,s)|ds; this means that 𝒜(𝒟)𝒟.

Step  2. We will show that the operator 𝒜 is weakly sequentially continuous.

Let (xn) be a sequence in 𝒟 and let (xn(t))x(t) in (E,w) for each tJ. Fix tJ. Since f satisfies assumptions (H1), we have f(t,xn(t)), converging weakly uniformly to f(t,x(t)). Hence the Lebesgue Dominated Convergence theorem for Pettis integral implies 𝒜xn(t) converging weakly uniformly to 𝒜x(t) in Eω. We do it for each tJ so 𝒜xn𝒜x. Then 𝒜:𝒟𝒟 is weakly sequentially continuous.

Step  3. The implication (2.3) holds. Now let V be a subset of 𝒟 such that Vconv¯(𝒜(V){0}). Clearly, V(t)conv¯(𝒜(V){0}) for all tJ. Hence, 𝒜V(t)𝒜𝒟(t),tJ, is bounded in E. Since function g is continuous on J, the set {g(t),tJ}¯E is compact, so β(g(t))=0. Using this fact, assumption (H4), Lemma 2.5 and the properties of the measure β, we have for each tJ(3.24)v(t)β(A(V)(t){0})β(A(V)(t))=β{0TG(t,s)f(s,V(s))ds}0T|G(t,s)|pf(s)β(V(s))dspfL0T|G(t,s)|v(s)dspfLvG*, which gives (3.25)vpfLvG*.

This means that (3.26)v[1-pfLG*]0. By (3.13) it follows that v=0, that is v(t)=0 for each tJ, and then V(t) is relatively weakly compact in E. In view of Lemma 2.9, we deduce that 𝒜 has a fixed point which is obviously a solution of the problem (1.1). This completes the proof.

4. An Example

In this section we give an example to illustrate the usefulness of our main result.

Example 4.1.

Let us consider the following fractional boundary value problem: (4.1)cDαu=219+etu1+u,tJ:=[0,T],1<α2,u(0)=λ1u(T)+μ1,u(0)=λ2u(T)+μ2.

Set T=1, f(t,u)=(2/(19+et))(u/(1+u)), (t,u)J×E, λ1=λ2=-1, μ1=μ2=0.

Clearly conditions (H1), (H2), and (H3) hold with pf(t)=2/(19+et). From (3.5), we have (4.2)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),if  0ts1.

We have (4.3)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),pfL=110.

A simple computation gives (4.4)G*<14Γ(α)+12Γ(α+1).

We shall check that condition (3.13) is satisfied. Indeed (4.5)pLG*<110[14Γ(α)+12Γ(α+1)]<1, which is satisfied for some α(1,2]. Then by Theorem 3.7, the problem (4.1) has at least one solution on J for values of α satisfying (4.5).

Acknowledgments

Wen-Xue Zhou’s work was supported by NNSF of China (11161027), NNSF of China (10901075), and the Key Project of Chinese Ministry of Education (210226).

Hilfer R. Applications of Fractional Calculus in Physics 2000 River Edge, NJ, USA World Scientific 10.1142/9789812817747 1890104 ZBL1235.74139 Sabatier J. Agrawal O. P. Tenreiro Machado J. A. Advances in Fractional Calculus 2007 Dordrecht, The Netherlands Springer 10.1007/978-1-4020-6042-7 2432163 ZBL1235.52015 Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier Science 2218073 10.1016/S0304-0208(06)80001-0 ZBL1206.26007 Lakshmikantham V. Leela S. Devi J. V. Theory of Fractional Dynamic Systems 2009 Cambridge, UK Cambridge Scientific Publishers Miller K. S. Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 New York, NY, USA John Wiley & Sons 1219954 ZBL0943.82582 Oldham K. B. Spanier J. The Fractional Calculus 1974 New York, NY, USA Academic Press 0361633 ZBL0292.26011 Podlubny I. Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press 1658022 ZBL1056.93542 Samko S. G. Kilbas A. A. Marichev O. I. Fractional Integrals and Derivatives 1993 Yverdon, Switzerland Gordon and Breach Science Publishers 1347689 ZBL0924.44003 Agarwal R. P. Benchohra M. Hamani S. Boundary value problems for differential inclusions with fractional order Advanced Studies in Contemporary Mathematics 2008 16 2 181 196 2404634 ZBL1152.26005 Ahmad B. Nieto J. J. Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations Abstract and Applied Analysis 2009 2009 9 494720 10.1155/2009/494720 2516016 ZBL1186.34009 Bai Z. H. Positive solutions for boundary value problem of nonlinear fractional differential equation Journal of Mathematical Analysis and Applications 2005 311 2 495 505 10.1016/j.jmaa.2005.02.052 2168413 ZBL1079.34048 El-Shahed M. Nieto J. J. Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order Computers & Mathematics with Applications 2010 59 11 3438 3443 10.1016/j.camwa.2010.03.031 2646314 ZBL1197.34003 Chang Y.-K. Nieto J. J. Some new existence results for fractional differential inclusions with boundary conditions Mathematical and Computer Modelling 2009 49 3-4 605 609 10.1016/j.mcm.2008.03.014 2483665 ZBL1165.34313 Li C. F. Luo X. N. Zhou Y. Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations Computers & Mathematics with Applications 2010 59 3 1363 1375 10.1016/j.camwa.2009.06.029 2579500 ZBL1189.34014 Jiao F. Zhou Y. Existence of solutions for a class of fractional boundary value problems via critical point theory Computers & Mathematics with Applications 2011 62 3 1181 1199 10.1016/j.camwa.2011.03.086 2824707 ZBL1235.34017 Wang J. R. Zhou Y. Analysis of nonlinear fractional control systems in Banach spaces Nonlinear Analysis 2011 74 17 5929 5942 10.1016/j.na.2011.05.059 2833364 ZBL1223.93059 Wang G. T. Ahmad B. Zhang L. H. Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order Nonlinear Analysis 2011 74 3 792 804 10.1016/j.na.2010.09.030 2738631 ZBL1214.34009 Yuan C. Multiple positive solutions for (n-1,1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations Electronic Journal of Qualitative Theory of Differential Equations 2010 36 1 12 2652066 Zhou W. Chu Y. Existence of solutions for fractional differential equations with multi-point boundary conditions Communications in Nonlinear Science and Numerical Simulation 2012 17 3 1142 1148 10.1016/j.cnsns.2011.07.019 2843780 Banaś J. Goebel K. Measures of Noncompactness in Banach Spaces 1980 New York, NY, USA Marcel Dekker 591679 Banaś J. Sadarangani K. On some measures of noncompactness in the space of continuous functions Nonlinear Analysis 2008 68 2 377 383 10.1016/j.na.2006.11.003 2369904 ZBL1134.46012 Guo D. Lakshmikantham V. Liu X. Nonlinear Integral Equations in Abstract Spaces 1996 373 Dordrecht, The Netherlands Kluwer Academic 1418859 Krzyśka S. Kubiaczyk I. On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces Demonstratio Mathematica 1999 32 2 323 330 1710255 ZBL0954.34050 Lakshmikantham V. Leela S. Nonlinear Differential Equations in Abstract Spaces 1981 2 Oxford, UK Pergamon Press 616449 Mönch H. Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces Nonlinear Analysis 1980 4 5 985 999 10.1016/0362-546X(80)90010-3 586861 ZBL0462.34041 O'Regan D. Fixed-point theory for weakly sequentially continuous mappings Mathematical and Computer Modelling 1998 27 5 1 14 10.1016/S0895-7177(98)00014-4 1616796 ZBL1185.34026 O'Regan D. Weak solutions of ordinary differential equations in Banach spaces Applied Mathematics Letters 1999 12 1 101 105 10.1016/S0893-9659(98)00133-5 1663477 ZBL0933.34068 Szufla S. On the application of measure of noncompactness to existence theorems Rendiconti del Seminario Matematico della Università di Padova 1986 75 1 14 847653 ZBL0589.45007 Szufla S. Szukała A. Existence theorems for weak solutions of nth order differential equations in Banach spaces Functiones et Approximatio Commentarii Mathematici 1998 26 313 319 Dedicated to Julian Musielak 1666630 Salem H. A. H. On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies Journal of Computational and Applied Mathematics 2009 224 2 565 572 10.1016/j.cam.2008.05.033 2492889 Salem H. A. H. El-Sayed A. M. A. Moustafa O. L. A note on the fractional calculus in Banach spaces Studia Scientiarum Mathematicarum Hungarica 2005 42 2 115 130 10.1556/SScMath.42.2005.2.1 2146147 ZBL1086.45004 Benchohra M. Graef J. R. Mostefai F.-Z. Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces Electronic Journal of Qualitative Theory of Differential Equations 2010 54 10 2684109 ZBL1206.26006 Pettis B. J. On integration in vector spaces Transactions of the American Mathematical Society 1938 44 2 277 304 10.2307/1989973 1501970 ZBL0019.41603 De Blasi F. S. On a property of the unit sphere in a Banach space 1977 21 3-4 259 262 0482402 ZBL0365.46015