1Department of Mathematics, Faculty of Science, Dohuk University, Kurdistan, Iraq 2Department of Mathematics, Faculty of Science, Zakho University, Kurdistan, Iraq 3Department of Mathematics and Science, College of Education and Basic Sciences, Ajman University of Science and Technology, UAE
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.
First of all, we recall some basic definitions; see [12–15].
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 .
Lemma 2.5. Let and , where , be a continuous function, then the solution of fractional differential equation (2.4) with the boundary condition (2.5) is
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
Therefore the solution of (2.4)-(2.5) is
which completes the proof.
3. The Main Result
Theorem 3.1. If the hypotheses (H1)–(H5) are satisfied, then the fractional integrodifferential equation (2.4)-(2.5) has a unique solution on .
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
where . 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.
Theorem 3.2. Assume that (H1)–(H5) hold with
Then the boundary value problem (2.4)-(2.5) has at least one element on .
Proof. Consider . We define the operators and as
Let us observe that if , then ,
where . 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
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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