1. Introduction

We investigate in this paper the existence and uniqueness of the mild solution for the fractional differential equation with infinite delay

(1.1)

where is a generator of an analytic semigroup on a Banach space such that for all and for every and . The function is continuous functions with additional assumptions.

The fractional derivative is understood here in the Caputo sense, that is,

(1.2)

where is called phase space to be defined in Section 2. For any function defined on and any , we denote by the element of defined by

(1.3)

The function represents the history of the state from up to the present time .

The theory of functional differential equations has emerged as an important branch of nonlinear analysis. It is worthwhile mentioning that several important problems of the theory of ordinary and delay differential equations lead to investigations of functional differential equations of various types (see the books by Hale and Verduyn Lunel [1], Wu [2], Liang et al. [3], Liang and Xiao [4–9], and the references therein). On the other hand the theory of fractional differential equations is also intensively studied and finds numerous applications in describing real world problems (see e.g., the monographs of Lakshmikantham et al. [10], Lakshmikantham [11], Lakshmikantham and Vatsala [12, 13], Podlubny [14], and the papers of Agarwal et al. [15], Benchohra et al. [16], Anguraj et al. [17], Mophou and N'Guérékata [18], Mophou et al. [19], Mophou and N'Guérékata [20], and the references therein).

Recently we studied in our paper [20] the existence of solutions to the fractional semilinear differential equation with nonlocal condition and delay-free

(1.4)

where is a positive real, is the generator of a -semigroup on a Banach space , with defined as above and

(1.5)

is a nonlinear function, is continuous, and . The derivative is understood here in the Riemann-Liouville sense.

In the present paper we deal with an infinite time delay. Note that in this case, the phase space plays a crucial role in the study of both qualitative and quantitative aspects of theory of functional equations. Its choice is determinant as can be seen in the important paper by Hale and Kato [21].

Similar works to the present paper include the paper by Benchohra et al. [16], where the authors studied an existence result related to the nonlinear functional differential equation

(1.6)

where is the standard Riemann-Liouville fractional derivative, in the phase space , with .

2. Preliminaries

From now on, we set . We denote by a Banach space with norm , the space of all -valued continuous functions on , and the Banach space of all linear and bounded operators on .

We assume that the phase space is a seminormed linear space of functions mapping into , and satisfying the following fundamental axioms due to Hale and Kato (see e.g., in [21]).

If , is continuous on and , then for every the following conditions hold:

  1. (i)

    is in ,

  2. (ii)
  3. (iii)

where is a constant, is continuous, is locally bounded, and , , are independent of .

For the function in , is a -valued continuous function on .

The space is complete.

Remark.

Condition (ii) in is equivalent to for all .

Let us recall some examples of phase spaces.

Example.

(E1) the Banach space of all bounded and uniformly continuous functions endowed with the supnorm.

(E2) the Banach space of all bounded and continuous functions such that endowed with the norm

(2.1)

() endowed with the norm

(2.2)

Note that the space is a uniform fading memory for .

Throughout this work will be a continuous function . Let be set defined by:

(2.3)

Remark.

We recall that the Cauchy Problem

(2.4)

where is a closed linear operator defined on a dense subset, is wellposed, and the unique solution is given by

(2.5)

where is a probability density function defined on such that its Laplace transform is given by

(2.6)

([22, cf. Theorem 2.1]).

Following [22, 23] we will introduce now the definition of mild solution to (1.1).

Definition 2.4.

A function is said to be a mild solution of (1.1) if satisfies

(2.7)

where

(2.8)

Remark 2.5.

Note that

(2.9)

since (cf. [23]).

3. Main Results

We present now our result.

Theorem 3.1.

Assume the following.

  • (H1) There exist such that for all ,

    (3.1)

  • (H2) There exists , with such that the function defined by:

(3.2)

satisfies for all . Here

(3.3)

Then (1.1) has a unique mild solution on .

Proof.

Consider the operator defined by

(3.4)

Let be the function defined by

(3.5)

Then . For each with , we denote by the function defined by

(3.6)

If verifies (2.7) then writing for , we have for and

(3.7)

Moreover .

Let

(3.8)

For any , we have

(3.9)

Thus is a Banach space. We define the operator by

(3.10)

It is clear that the operator has a unique fixed point if and only if has a unique fixed point. So let us prove that has a unique fixed point. Observe first that is obviously well defined. Now, consider . For any , we have

(3.11)

So using , (2.9) and (3.3), we obtain for all

(3.12)

which according to gives

(3.13)

Therefore

(3.14)

And since , we conclude by way of the Banach's contraction mapping principle that has a unique fixed point . This means that has a unique fixed point which is obviously a mild solution of (1.1) on .

4. Application

To illustrate our result, we consider the following Lotka-Volterra model with diffusion:

(4.1)

where and is a positive function on with .

Now let and consider the operator defined by

(4.2)

Clearly is dense in .

Define

(4.3)

We choose as in Example (E3) above. Put

(4.4)

Then we get

(4.5)

where is obviously Lipschitzian in uniformly in . Thus we can state what follows.

Theorem 4.1.

Under the above assumptions (4.1) has a unique mild solution.