A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay
© G. M. Mophou and G. M. N'Guérékata. 2010
Received: 28 November 2009
Accepted: 21 January 2010
Published: 26 January 2010
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.
We investigate in this paper the existence and uniqueness of the mild solution for the fractional differential equation with infinite delay
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 , Wu , Liang et al. , 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. , Lakshmikantham , Lakshmikantham and Vatsala [12, 13], Podlubny , and the papers of Agarwal et al. , Benchohra et al. , Anguraj et al. , Mophou and N'Guérékata , Mophou et al. , Mophou and N'Guérékata , and the references therein).
Recently we studied in our paper  the existence of solutions to the fractional semilinear differential equation with nonlocal condition and delay-free
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 .
Similar works to the present paper include the paper by Benchohra et al. , where the authors studied an existence result related to the nonlinear functional differential equation
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 ).
Let us recall some examples of phase spaces.
([22, cf. Theorem 2.1]).
since (cf. ).
3. Main Results
We present now our result.
To illustrate our result, we consider the following Lotka-Volterra model with diffusion:
Then we get
Under the above assumptions (4.1) has a unique mild solution.
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