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Existence and Uniqueness Theorem of Fractional Mixed Volterra-Fredholm Integrodifferential Equation with Integral Boundary Conditions
We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.
In the last century, notable contributions have been made to both the theory and applications of the fractional differential equations. For the theory part, Momani and Hadid have investigated the local and global existence theorem of both fractional differential equation and fractional integrodifferential equations; see [1–6]. Fractional-order differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.
Integrodifferential equations with integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics and cellular systems. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper . In , Tidke studied the problem of existence of global solutions to nonlinear mixed Volttera-Fredholm integrodifferential equations with nonlocal condition.
Ahmad and Nieto  studied some existence results for boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral equation.
Very recently N’Guérékata  discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. Anguraj et al.  studied the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition.
Motivated by these works, we study in this paper the existence of solution of boundary value problem for fractional integrodifferential equations ( in the case ) in Banach spaces by using Banach and Krasnosel'skii fixed-point theorems.
Definition 2.1. For a function given on the interval , the Caputo fractional order derivative of is defined by where and denotes the integer part of .
Lemma 2.2. Let , then for some , .
Definition 2.3. Let be a function which is defined almost everywhere (a.e) on , for , we define provided that the integral (Lebesgue) exists.
Theorem 2.4 (Krasnosel’skii fixed point theorem). Let be a closed-convex bounded nonempty subset of a Banach space . Let and be two operators such that , whenever , is compact and continuous; is a contraction mapping,then there exists such that .
Let be a Banach space with the norm . Let be Banach space of all continuous functions , with supermum norm . Consider the fractional mixed Volttera-Fredholm integrodifferential equation with boundary conditions, which has the form where , is the Caputo fractional derivative and the nonlinear functions , and satisfy the following hypotheses: there exists constants such that , for ,there exists constants such that and There exists continuous functions and such that and , for every and ,there exists continuous functions and such that and there exists continuous function , and is positive constant such that and , for every and , where is continuous nondecreasing function satisfying , where is a continuous function .
Proof. By Lemma 2.2, we reduce the problem (2.4)-(2.5) to an equivalent integral equation In view of the relations and , for , we obtain Applying the boundary condition (2.5), we find that that is, Therefore the solution of (2.4)-(2.5) is which completes the proof.
3. The Main Result
Proof. Define by
We show that has a fixed point on Br. This fixed point is then a solution of (2.4)-(2.5). Firstly, we show that , where . For , we have
Since we have , and , we get
Now, take and for each , we obtain by using (H1)–(H5), we get Since we have , , and, Let , , then where .
As , therefore is a contraction. Thus, the conclusion of the theorem is followed by the contraction mapping principle.
Proof. Consider . We define the operators and as
Let us observe that if , then ,
Now we prove that is contraction mapping, Let , we obtain It is clear that is contraction mapping, since is continuous, then is continuous Hence, is uniformly bounded on . Now, let us prove that is equicontinuous, let and . Using the fact that is bounded on the compact set , thus ,we get
So is relatively compact. By Arzela-Ascoli theorem, is compact. Now we conclude the result of the theorem of Krasnosel’skii theorem.
Example 3.3. Consider the following fractional mixed Volterra-Fredholm integrodifferential equation: with integral boundary conditions Here, Hence, the conditions (H1)–(H5) hold with , , , , , , , and , thus We conclude from the above example that the integrodifferential equation has unique solution.
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