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A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay
Advances in Difference Equations volume 2010, Article number: 674630 (2010)
Abstract
We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: , , , , with and . We prove the existence (and uniqueness) of solutions, assuming that is a linear closed operator which generates an analytic semigroup on a Banach space by means of the Banach's fixed point theorem. This generalizes some recent results.
1. Introduction
We investigate in this paper the existence and uniqueness of the mild solution for the fractional differential equation with infinite delay
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,
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
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 delayfree
where is a positive real, is the generator of a semigroup on a Banach space , with defined as above and
is a nonlinear function, is continuous, and . The derivative is understood here in the RiemannLiouville 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
where is the standard RiemannLiouville 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:

(i)
is in ,
 (ii)
 (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
() endowed with the norm
Note that the space is a uniform fading memory for .
Throughout this work will be a continuous function . Let be set defined by:
Remark.
We recall that the Cauchy Problem
where is a closed linear operator defined on a dense subset, is wellposed, and the unique solution is given by
where is a probability density function defined on such that its Laplace transform is given by
([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
where
Remark 2.5.
Note that
since (cf. [23]).
3. Main Results
We present now our result.
Theorem 3.1.
Assume the following.

(H_{1}) There exist such that for all ,
(3.1) 
(H_{2}) There exists , with such that the function defined by:
satisfies for all . Here
Then (1.1) has a unique mild solution on .
Proof.
Consider the operator defined by
Let be the function defined by
Then . For each with , we denote by the function defined by
If verifies (2.7) then writing for , we have for and
Moreover .
Let
For any , we have
Thus is a Banach space. We define the operator by
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
So using , (2.9) and (3.3), we obtain for all
which according to gives
Therefore
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 LotkaVolterra model with diffusion:
where and is a positive function on with .
Now let and consider the operator defined by
Clearly is dense in .
Define
We choose as in Example (E3) above. Put
Then we get
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.
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Keywords
 Banach Space
 Phase Space
 Functional Equation
 Probability Density Function
 Kato