Existence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditions

In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. As long as the Banach space is reflexive, theweak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [1, 2]. De Blasi [3] introduced the concept of measure of weak noncompactness and proved the analogue of Sadovskiis fixed point theorem for the weak topology (see also [4]). As stressed in [5], in many applications, it is always not possible to show the weak continuity of the involved mappings, while the sequential weak continuity offers no problem. This is mainly due to the fact that Lebesgues dominated convergence theorem is valid for sequences but not for nets. Recall that a mapping between two Banach spaces is sequentially weakly continuous if it maps weakly convergent sequences into weakly convergent sequences. The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored. There are many papers dealing with multipoint boundary value problems both on resonance case and on nonresonance case; for more details see [6–11]. However, as far as we know, few results can be found in the literature concerning multipoint boundary value problems for fractional differential equations in Banach spaces andweak topologies. Zhou et al. [12] discuss the existence of solutions for nonlinear multipoint boundary value problem of integrodifferential equations of fractional order as follows: cD0+x (t) = f (t, x (t) , (Hx) (t) , (Kx) (t)) , t ∈ [0, 1] , α ∈ (1, 2] , a1x (0) − b1x󸀠 (0) = d1x (ξ1) , a2x (1) + b2x󸀠 (1) = d2x (ξ2) , (1)


Introduction
In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology.As long as the Banach space is reflexive, the weak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [1,2].De Blasi [3] introduced the concept of measure of weak noncompactness and proved the analogue of Sadovskiis fixed point theorem for the weak topology (see also [4]).As stressed in [5], in many applications, it is always not possible to show the weak continuity of the involved mappings, while the sequential weak continuity offers no problem.This is mainly due to the fact that Lebesgues dominated convergence theorem is valid for sequences but not for nets.Recall that a mapping between two Banach spaces is sequentially weakly continuous if it maps weakly convergent sequences into weakly convergent sequences.
The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.There are many papers dealing with multipoint boundary value problems both on resonance case and on nonresonance case; for more details see [6][7][8][9][10][11].However, as far as we know, few results can be found in the literature concerning multipoint boundary value problems for fractional differential equations in Banach spaces and weak topologies.Zhou et al. [12] discuss the existence of solutions for nonlinear multipoint boundary value problem of integrodifferential equations of fractional order as follows: 0+  () =  (,  () , () () , () ()) ,  ∈ [0, 1] ,  ∈ (1,2] ,  1  (0) −  1   (0) =  1  ( 1 ) ,  2  (1) +  2   (1) =  2  ( 2 ) , (2) Moreover, theory for boundary value problem of integrodifferential equations of fractional order in Banach spaces endowed with its weak topology has been few studied until now.In [13], we discussed the existence theorem of weak 2 International Journal of Differential Equations solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces : and obtain a new result by using the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals, where    0+ denotes the fractional Caputo derivative and the operators given by Our analysis relies on the Krasnoselskii fixed point theorem combined with the technique of measure of weak noncompactness.
The problems of our research are different between this paper and paper [13].In paper [13], we studied two point boundary value problem by using the corresponding Green's function and fixed point theorems; moreover, we get some good results.In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology.Our results generalized some classical results and improve the assumptions conditions, so our results improve the results in [13].
The paper is organized as follows: In Section 2 we recall some basic known results.In Section 3 we discuss the existence theorem of weak solutions for problem (5).

Preliminaries
Throughout this paper, we introduce notations, definitions, and preliminary results which will be used.
Let Ω  be the collection of all nonempty bounded subsets of , and let W  be the subset of Ω  consisting of all weakly compact subsets of .Let   denote the closed ball in  centered at 0 with radius  > 0. The De Blasi [14] measure of weak noncompactness is the map  : Ω  → [0, ∞) defined by for all  ∈ Ω  .The fundamental tool in this paper is the measure of weak noncompactness; for some properties of () and more details see [3].Now, for the convenience of the reader, we recall some useful definitions of integrals.
Lemma 8 (see [17]).Let  ⊂ (,) be bounded and equicontinuous.Then (()) is continuous on  and We give the fixed point theorem, which play a key role in the proof of our main results.
Lemma 9 (see [20]).Let  be a Banach space and  a regular and set additive measure of weak noncompactness on .Let  be a nonempty closed convex subset of ,  0 ∈ , and  0 a positive integer.Suppose  :  →  is -convex power condensing about  0 and  0 .If  is weakly sequentially continuous and () is bounded, then  has a fixed point in .
The following we recall the definition of the Caputo derivative of fractional order.Definition 10.Let  :  →  be a function.The fractional HKP-integral of the function  of order  ∈ R + is defined by In the above definition the sign "∫" denotes the HKPintegral integral.
Definition 11.The Riemann-Liouville derivative of order  with the lower limit zero for a function  : [0, ∞) →  can be written as Definition 12.The Caputo fractional derivative of order  for a function  : [0, ∞) →  can be written as where  = [] + 1 and [] denotes the integer part of .

Main Results
In this section, we present the existence of solutions to problem (5) in the space (,   ).
The following we give the corresponding Greens function for problem (5).Lemma 6.Let Δ ̸ = 0,  ∈ (,   ) and  ∈ (1, 2], then the unique solution of is given by where the Green function  is given by Proof.Based on the idea of paper [7], assuming that () satisfies ( 18), by Lemma 15, we formally put for some constants  1 ,  2 ∈ R.

International Journal of Differential Equations
We will show that  satisfies the assumptions of Lemma 8; the proof will be given in three steps.
Step 1.We shall show that the operator  maps into itself.
and this estimation shows that  maps  into itself.
Step 2. We will show that the operator  is weakly sequentially continuous.In order to be simple, we denote () = ()() = ∫  (42) Therefore, the operators ,  are weakly sequentially continuous in .