Existence Results of Fractional Mixed Volterra -fredholm Integrodifferential Equations with Integral Boundary Conditions

In this paper, we give an existence results for the fractional mixed Volterra-Fredholm integrodifferential equation with Integral Boundary Conditions. First, we use the Banach contraction principle to prove the existence and the uniqueness of the solution for the boundary value problem. Second, we study the existence of the solutions for the boundary value problem. using Krasnoselskii's fixed point theorem.


Introduction
We consider the following fractional integrodifferential boundary value problem with integral boundary condition (1.1) where is the standard Caputo derivative, and and , the Banach space with norm: and the functions are continuous functions. Here, For brevity let The problem of existence and uniqueness of solution for fractional differential equations have been considered by many authors; see for example [1], [2], [3], [7], [8], [9], [11], [14]. An integrodifferential equation is an equation which involves both integrals and derivatives of an unknown function. The existence and uniqueness problems of fractional nonlinear differential and integrodifferential equations as a basic theoretical part of some applications are investigated by many authors (see for examples [1], [13], and [14]). It arises in many fields like electronic, fluid dynamics, biological models, and chemical kinetics. A well-known example is the equations of basic electric circuit analysis. In recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established; see for example [10], [12] .
In [4] the authors studied the non-fractional mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions using Leray -Schauder theorem, and In [15] the authors studied the fractional differential equations with integral boundary conditions By means of the famous Banach contraction mapping principle.
we study in this paper the existence of the solution of the boundary value problem for fractional integrodifferential equations (in the case of ) with integral boundary conditions in Banach spaces by using Banach and Krasnoselskii's fixed point theorems.
For the sake of clarity, we list the necessary definitions from fractional calculus theory here. These definitions can be found in the recent literature.

Definition 2.1 [5] Let
, for a function . The the fractional integral of order of is defined by provided the integral exists.

Definition 2.2 The Caputo derivative of a function
is given by provided the right side is point wise defined on ,where , and denotes the integer part of the real number .
The properties of the above operators can be found in [6] and the general theory of fractional differential equations can be found in [5]. denotes the Gamma function: The Gamma function satisfies the following basic properties: To proceed, we need the following assumptions:

Proof . By (H2) we have
Similarly, for the other estimate, we use assumption (H2) ,to get

Main Results
In this section, we give the existence and uniqueness of the solutions for problem (1.1). Since for the same reason as the operator is also continuous, it suffices to prove that is uniformly bounded and is compact to prove that is compact. Let . then from which we deduce that is uniformly bounded on .
Then on the other hand, for , which does not depend on y. So is relatively compact. By the Arzela-Ascoli Theorem,U is compact. Thus, we have proved that is continuous and compact, is a contraction mapping and if . Hence, the Krasnoselskii theorem lead us to conclude that the boundary value problems (1.1) has at least one solution on .

An Example
In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional BVP Hence for all Therefore Then by Theorem 3.1 the fractional BVP (4.1) has a unique solution on .