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A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay

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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 [49], 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


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 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


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.


Condition (ii) in is equivalent to for all .

Let us recall some examples of phase spaces.


(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:



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




Remark 2.5.

Note that


since (cf. [23]).

3. Main Results

We present now our result.

Theorem 3.1.

Assume the following.

  • (H1) There exist such that for all ,

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


satisfies for all . Here


Then (1.1) has a unique mild solution on .


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 .



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




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:


where and is a positive function on with .

Now let and consider the operator defined by


Clearly is dense in .



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.


  1. 1.

    Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.

  2. 2.

    Wu J: Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996:x+429.

  3. 3.

    Liang J, Huang F, Xiao T: Exponential stability for abstract linear autonomous functional-differential equations with infinite delay. International Journal of Mathematics and Mathematical Sciences 1998,21(2):255-259. 10.1155/S0161171298000362

  4. 4.

    Liang J, Xiao TJ: Functional-differential equations with infinite delay in Banach spaces. International Journal of Mathematics and Mathematical Sciences 1991,14(3):497-508. 10.1155/S0161171291000686

  5. 5.

    Liang J, Xiao T-J: The Cauchy problem for nonlinear abstract functional differential equations with infinite delay. Computers & Mathematics with Applications 2000,40(6-7):693-703. 10.1016/S0898-1221(00)00188-7

  6. 6.

    Liang J, Xiao T-J: Solvability of the Cauchy problem for infinite delay equations. Nonlinear Analysis: Theory, Methods & Applications 2004,58(3-4):271-297. 10.1016/

  7. 7.

    Liang J, Xiao T-J: Solutions to nonautonomous abstract functional equations with infinite delay. Taiwanese Journal of Mathematics 2006,10(1):163-172.

  8. 8.

    Liang J, Xiao T-J, van Casteren J: A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Applied Mathematics Letters 2004,17(4):473-477. 10.1016/S0893-9659(04)90092-4

  9. 9.

    Xiao T-J, Liang J: Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1442-e1447. 10.1016/

  10. 10.

    Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge, UK; 2009.

  11. 11.

    Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337-3343. 10.1016/

  12. 12.

    Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677-2682. 10.1016/

  13. 13.

    Lakshmikantham V, Vatsala AS: Theory of fractional differential inequalities and applications. Communications in Applied Analysis 2007,11(3-4):395-402.

  14. 14.

    Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.

  15. 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: 1-21.

  16. 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):1340-1350. 10.1016/j.jmaa.2007.06.021

  17. 17.

    Anguraj A, Karthikeyan P, N'Guérékata GM: Nonlocal Cauchy problem for some fractional abstract integro-differential equations in Banach spaces. Communications in Mathematical Analysis 2009,6(1):31-35.

  18. 18.

    Mophou GM, N'Guérékata GM: Mild solutions for semilinear fractional differential equations. Electronic Journal of Differential Equations 2009,2009(21):1-9.

  19. 19.

    Mophou GM, Nakoulima O, N'Guérékata GM: Existence results for some fractional differential equations with nonlocal conditions. Nonlinear Studies 2010,17(1):15-22.

  20. 20.

    Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315-322. 10.1007/s00233-008-9117-x

  21. 21.

    Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11-41.

  22. 22.

    El-Borai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons and Fractals 2002,14(3):433-440. 10.1016/S0960-0779(01)00208-9

  23. 23.

    El-Borai MM: On some stochastic fractional integro-differential equations. Advances in Dynamical Systems and Applications 2006,1(1):49-57.

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Correspondence to Gisle M. Mophou.

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  • Banach Space
  • Phase Space
  • Functional Equation
  • Probability Density Function
  • Kato