 Research Article
 Open Access
 Published:
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.
References
 1.
Hale JK, Verduyn Lunel SM: Introduction to FunctionalDifferential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.
 2.
Wu J: Theory and Applications of Partial FunctionalDifferential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996:x+429.
 3.
Liang J, Huang F, Xiao T: Exponential stability for abstract linear autonomous functionaldifferential equations with infinite delay. International Journal of Mathematics and Mathematical Sciences 1998,21(2):255259. 10.1155/S0161171298000362
 4.
Liang J, Xiao TJ: Functionaldifferential equations with infinite delay in Banach spaces. International Journal of Mathematics and Mathematical Sciences 1991,14(3):497508. 10.1155/S0161171291000686
 5.
Liang J, Xiao TJ: The Cauchy problem for nonlinear abstract functional differential equations with infinite delay. Computers & Mathematics with Applications 2000,40(67):693703. 10.1016/S08981221(00)001887
 6.
Liang J, Xiao TJ: Solvability of the Cauchy problem for infinite delay equations. Nonlinear Analysis: Theory, Methods & Applications 2004,58(34):271297. 10.1016/j.na.2004.05.005
 7.
Liang J, Xiao TJ: Solutions to nonautonomous abstract functional equations with infinite delay. Taiwanese Journal of Mathematics 2006,10(1):163172.
 8.
Liang J, Xiao TJ, van Casteren J: A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Applied Mathematics Letters 2004,17(4):473477. 10.1016/S08939659(04)900924
 9.
Xiao TJ, Liang J: Blowup and global existence of solutions to integral equations with infinite delay in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1442e1447. 10.1016/j.na.2009.01.204
 10.
Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge, UK; 2009.
 11.
Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):33373343. 10.1016/j.na.2007.09.025
 12.
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):26772682. 10.1016/j.na.2007.08.042
 13.
Lakshmikantham V, Vatsala AS: Theory of fractional differential inequalities and applications. Communications in Applied Analysis 2007,11(34):395402.
 14.
Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.
 15.
Agarwal RP, Benchohra M, Slimani BA: Existence results for differential equations with fractional order and impulses. Memoirs on Differential Equations and Mathematical Physics 2008, 44: 121.
 16.
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008,338(2):13401350. 10.1016/j.jmaa.2007.06.021
 17.
Anguraj A, Karthikeyan P, N'Guérékata GM: Nonlocal Cauchy problem for some fractional abstract integrodifferential equations in Banach spaces. Communications in Mathematical Analysis 2009,6(1):3135.
 18.
Mophou GM, N'Guérékata GM: Mild solutions for semilinear fractional differential equations. Electronic Journal of Differential Equations 2009,2009(21):19.
 19.
Mophou GM, Nakoulima O, N'Guérékata GM: Existence results for some fractional differential equations with nonlocal conditions. Nonlinear Studies 2010,17(1):1522.
 20.
Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315322. 10.1007/s002330089117x
 21.
Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):1141.
 22.
ElBorai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons and Fractals 2002,14(3):433440. 10.1016/S09600779(01)002089
 23.
ElBorai MM: On some stochastic fractional integrodifferential equations. Advances in Dynamical Systems and Applications 2006,1(1):4957.
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Mophou, G.M., N'Guérékata, G.M. A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay. Adv Differ Equ 2010, 674630 (2010). https://doi.org/10.1155/2010/674630
Received:
Accepted:
Published:
Keywords
 Banach Space
 Phase Space
 Functional Equation
 Probability Density Function
 Kato