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

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.

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].

Recently Mophou and N'Guérékata [17], studied the Cauchy problem with nonlocal conditions

(1.1)

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.

Motivated by the above mentioned works in this manuscript we discuss the existence and the uniqueness of the solution for the following fractional integrodifferential equation with nonlocal integral initial condition in Banach Space:

(1.2)

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 .

For the sake of the shortness let

(A)

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.

The fractional (arbitrary) order Riemann-Liouville integral (on the right and on the left) of the function of order is defined by

(2.1)

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.

The Riemann-Liouville fractional integral operator of order , of a function is defined as

(2.2)

Definition 2.4.

The Caputo's derivative of fractional order for a suitable function is defined by

(2.3)

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.

Let and . Then

(2.4)

Lemma 2.6.

If for , and if satisfies for and , then

(2.5)

Proof.

A direct computation shows

(2.6)

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

We assume the following.

• (A1) If and a nonnegative, bounded , there exist , for such that

  

  (3.1)

• (A2) There exist positive constants , , and such that

  

  (3.2)

• for all , , and .

• (A3) There exist positive constants , , and such that

  

  (3.3)

• for all , , and .

• (A4) There exist positive constants , , and such that

  

  (3.4)

• 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.

If (A1) holds with , then the problem (1.2) is equivalent to the following equation:

(3.5)

Proof.

By Lemma 2.5 and (1.2), we have

(3.6)

Therefore,

(3.7)

So,

(3.8)

and then

(3.9)

Conversely, if is a solution of (3.5), then for every , according to Definition 2.4 we have

(3.10)

It is obvious that . This completes the proof.

Lemma 3.2.

If (A3) and (A4) are satisfied, , are defined in (A), then the conditions

(3.11)

are satisfied for any , and .

Proof.

By (A3), we have

(3.12)

On the other hand,

(3.13)

Similarly, for the other conditions, we use assumption (A4), to get

(3.14)

Theorem 3.3.

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

Proof.

We use the Banach contraction principle to prove the existence and uniqueness of the solution to (1.2). Let , where and define the operator on the Banach space by

(3.15)

Firstly, we show that the operator maps into itself. By using (A1) and triangle inequality, we have

(3.16)

Now, if (A2) is satisfied, then

(3.17)

Using Lemma 3.2 and (A3), we have

(3.18)

if , we have

(3.19)

Thus . Next, we prove that is a contraction mapping. For this, let . Applying (A2), we have

(3.20)

then using (A3), (A4) and Lemma 3.2, one gets

(3.21)

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.

Choose and consider . Now define on the operators by

(3.22)

Let us observe that if then . Indeed it is easy to check the inequality

(3.23)

We can easily show that that is a contraction mapping. Let . Then

(3.24)

where depends only on the parameter of the problem and hence is contraction. Since is continuous, then is continuous in view of (A1). Let us now note that is uniformly bounded on . This follows from the inequality

(3.25)

Now let us prove that is equicontinuous.

Let and . Using the fact that is bounded on the compact set (thus , we will get

(3.26)

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

Consider the following fractional integrodifferential equation:

(4.1)

where . Take . Set , , , . Then it is clear that

(4.2)

So, (A1) is satisfied. Let and . Then we have

(4.3)

Hence the conditions (A1)–(A4) hold with , , , . Choose and . Indeed

(4.4)

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].