- Research Article
- Open Access

# Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

- A. Anguraj
^{1}, - P. Karthikeyan
^{2}and - J. J. Trujillo
^{3}Email author

**2011**:690653

https://doi.org/10.1155/2011/690653

© A. Anguraj et al. 2011

**Received:**14 November 2010**Accepted:**6 January 2011**Published:**11 January 2011

## Abstract

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition , where , , and is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.

## Keywords

- Banach Space
- Fractional Order
- Fractional Differential Equation
- Functional Differential Equation
- Integrodifferential Equation

## 1. Introduction

Recently it have been proved that the differential models involving nonlocal derivatives of fractional order arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in many fields, for instance, physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth (see [1–6]). In fact, such models can be considered as an efficient alternative to the classical nonlinear differential models to simulate many complex processes (see [7]). For instance, fractional differential equations are an excellent tool to describe hereditary properties of viscoelastic materials and, in general, to simulate the dynamics of many processes on anomalous media. Theory of fractional differential equations has been extensively studied by several authors as Delbosco and Rodino [8], Kilbas et al. [6], Lakshmikantham et al. [9–11], and also see [2, 12–16].

in general Banach space with , and is the infinitesimal generator of a -semigroup of bounded linear operators. By means of the Krasnoselskii's Theorem, existence of solutions was also obtained.

Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces.

Very recently N'Guérékata [2, 18] discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. The nonlocal Cauchy problem is discussed by authors in [15] using the fixed-point concepts. Tidke [19] studied the nonfractional mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions using Leray-Schauder Theorem.

where , , , , is a continuous function on with values in the Banach space and , and , , and are continuous -valued functions. Here , and . The operator denotes the Caputo fractional derivative of order .

The paper is organized as follows. In Section 2, some definitions, lemmas and preliminary results are introduced to be used in the sequel. Section 3 will involve the assumptions, main results and proofs of existence problem of (1.2), together with a nonlocal initial condition. Finally an example is presented.

## 2. Preliminaries

Let be a real Banach space and the zero element of . Let be the Banach space of measurable functions which are Lebesgue integrable, equipped with the norm . We will use the following notation and . A function is called a solution of (1.2) if it satisfies (1.2).

Definition 2.1.

A real function
is said to be in the space
if there exists a real number
, such that
, where
, while
is said to be in the space
if and only if
*.*

Definition 2.2.

where is the Gamma function of Euler.

When , we write , where for , for , and represents the Convolution of Laplace. Then, it is well known that as , where is the Delta function.

Definition 2.3.

Definition 2.4.

where denotes the integer part of real number .

It is obvious that the Caputo's derivative of a constant is equal to 0.

Lemma 2.5.

Lemma 2.6.

Proof.

Theorem 2.7 (Krasnoselkii).

Let be a Banach space, let be a bounded closed convex subset of and let , be maps of into such that for every pair . If A is completely continuous and B is a contraction then the equation has a solution on S.

## 3. Main Results

- (A1)
- (A2)
- for
all , , and .

- (A3)
- for
all , , and .

- (A4)
- for
all , , and .

- (A5)
is such that .

Firstly, we obtain the following lemmas to prove the main results on the existence of solutions to (1.2).

Lemma 3.1.

Proof.

It is obvious that . This completes the proof.

Lemma 3.2.

are satisfied for any , and .

Proof.

Theorem 3.3.

If (A1)–(A5) are satisfied, then the fractional integrodifferential equation (1.2) has a unique solution continuous in .

Proof.

where = depends on the parameter of the problem. Therefore has a unique fixed-point , which is a solution of (3.5), and hence is a solution of (1.2).

Theorem 3.4.

Assume (A1)–(A4) holds. If , then (1.2) has at least one solution on .

Proof.

Now let us prove that is equicontinuous.

which does not depend on . So is relatively compact. By the Arzela-Ascoli Theorem, is compact. We now conclude the result of the theorem based on the Krasnoselkii's theorem above.

## 4. Example

Further (A5) is satisfied by a suitable choice of . Then by Lemma 3.2 the problem (1.2) has a unique solution on [0,1].

## Declarations

### Acknowledgment

This paper has been partially supported by MICINN (project MTM2010-16499) to which the authors are very thankful.

## Authors’ Affiliations

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