On the existence of solutions for fractional boundary valued problems with integral boundary conditions involving measure of non compactness

Q uite recently, fractional differential equations become one of the most important research topic, since their applications in various applied science, as in physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, for example see [1,2]. many results were given concerning the existence and uniqueness of the solution of such equations by using various techniques, while the fixed point theory tool still one of the efficacy methods, see for example. Several authors tried to develop a technique that depends on the Darbo or the Monch fixed point theorems with the Hausdorff or Kuratowski measure of noncompactness. This article deals with the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral boundary conditions. We furnish an example to illustrate our results. Consider the following boundary value problems:  cDαx(t) + f (t, x(t),c Dαx(t)) = 0, 0 ≤ t ≤ 1, 1 < α ≤ 2 ax(0)− bx′(0) = 0 x(1) = ∫ 1 0 g(s, x(s))ds + λ (1)


Introduction
Q uite recently, fractional differential equations become one of the most important research topic, since their applications in various applied science, as in physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, for example see [1,2]. many results were given concerning the existence and uniqueness of the solution of such equations by using various techniques, while the fixed point theory tool still one of the efficacy methods, see for example. Several authors tried to develop a technique that depends on the Darbo or the Monch fixed point theorems with the Hausdorff or Kuratowski measure of noncompactness.
This article deals with the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral boundary conditions. We furnish an example to illustrate our results.
Consider the following boundary value problems: where λ > 0, c D α , 1 < α ≤ 2 is the Caputo fractional derivative, f and g are given functions f : Let L 1 ([0; 1], R) be the Banach space of measurable functions x : [0, 1] −→ R which are Bochner integrable, equipped with the norm Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness. Definition 1. [3,4] Let E be a Banach space and Ω E be the bounded subsets of E. The Kuratowski measure of noncompactness is the map α : The Kuratowski measure of noncompactness satisfies the following properties; Lemma 1. [3,4] Let A and B bounded sets. Then

Definition 2.
[5] The Riemann Liouville fractional integral of order q > 0 of x : (0, ∞) → R is given by provided that the right hand side is defined on (0, ∞).

Theorem 1.
[1] Let X be a Banach space and 0 ∈ C be a nonempty, bounded, closed and convex subset of X. Suppose a continuous mapping N : C → C is such that for all non empty subsets V of C, where 0 ≤ k < 1, and µ is the Kuratowski measure of noncompactness, then N has a fixed point in C.

Theorem 2.
[6] Let C be a bounded, closed and convex subset of a Banach space such that 0 ∈ C, and let T be a continuous mapping of C into itself. If the implication holds for every subset V of C, then T has a fixed point.

Lemma 4.
[7] Let D be a bounded, closed and convex subset of the Banach space C(J, X), G a continuous function on J × J and f a function from J × X → X which satisfies the Carathéodory conditions, and suppose there exists p ∈ L 1 (J, R + ) such that, for each t ∈ J and each bounded set B ⊂ X, we have Lemma 5. Let X be a Banach space and F ⊂ C(J, X). If the following conditions are satisfied: • family F in C(J, X) is called uniformly bounded if there exists a positive constant K such that | f (t)| ≤ K for all t ∈ J and all f ∈ F;

Main results
Lemma 6. Let 1 < α < 2 and y ∈ C([0, 1]). A function x is a solution of the fractional integral equation where G is the Green function given by if and only if x is a solution of the fractional boundary value problem Proof. By Lemma 3, we reduce (4)) to an equivalent integral equation for some constants c 0 , c 1 ∈ X. Boundary conditions of (4) give The proof is complete.
Assume that (A1): There exist K > 0 and L > 0 such that (A2): For any bounded subset A and B of X we have or µ is a measure of non-compactness. (A3): There exists N > 0 such that g(t, x(t)) − g(t, y(t) ≤ N x − y .

Theorem 3. Under the assumptions (A1)-(A5) the Problem (1) has a solution provided
Proof. Transform the Problem (1) into a fixed point problem. Consider the operator T : Clearly, the fixed points of the operator T are solutions of the Problem (1). We shall show that T satisfies the assumptions of Theorem 3. The proof will be given in three steps.
Therefore, T is continuous. Now, let It is clear that B r is a bounded, closed and convex subset of X.
According to (5) and (6), we have As t 2 −→ t 1 , the right hand side of the above inequality tends to zero, so T is equicontinuous. Let V ⊂ TB r , such as V = {Tx, x ∈ B r }, so V ⊂ conv(T(V) ∪ {0}). The subset V is bounded and equicontinuous, so the function t : −→ µ(V(t)) ∈ R is continuous on [0, 1]. Using Lemma 4 and the properties of the measure µ, we have, Thus for each s ∈ [0, 1], and µ(g(t, V(t)) = µ g(t, x(t), x(t) ∈ V(t) ≤ Nv(t). Then which gives v ∞ = 0, that is to say v(t) = 0, for each t ∈ [0, 1], then V(t) is relatively compact in X. In view of the lemma of Ascoli -Arzela, V is relatively compact in B r . Applying now the Theorem of Mönch, we conclude that T has a fixed point which is a solution of the Problem (1). This completes the proof.