Advances in Difference Equations
Volume 2010 (2010), Article ID 674630, 8 pages
doi:10.1155/2010/674630
Research Article

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

Gisle M. Mophou1 and Gaston M. N'Guérékata2

1Département de Mathématiques et Informatique, Université des Antilles et de La Guyane, Campus Fouillole 97159 Pointe-à-Pitre Guadeloupe (FWI), France
2Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

Received 28 November 2009; Accepted 21 January 2010

Academic Editor: A. Pankov

Copyright © 2010 Gisle M. Mophou and Gaston M. N'Guérékata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: 𝐷 𝛼 𝑥 ( 𝑡 ) + 𝐴 𝑥 ( 𝑡 ) = 𝑓 ( 𝑡 , 𝑥 𝑡 ) , 𝑡 ∈ [ 0 , 𝑇 ] , 𝑥 ( 𝑡 ) = 𝜙 ( 𝑡 ) , 𝑡 ∈ ] − ∞ , 0 ] , with 𝑇 > 0 and 0 < 𝛼 < 1 . We prove the existence (and uniqueness) of solutions, assuming that − 𝐴 is a linear closed operator which generates an analytic semigroup ( 𝑇 ( 𝑡 ) ) 𝑡 ≥ 0 on a Banach space 𝕏 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

𝐷 𝛼 𝑥 ( 𝑡 ) + 𝐴 𝑥 ( 𝑡 ) = 𝑓 𝑡 , 𝑥 𝑡 [ ] , ] ] , , 𝑡 ∈ 𝐼 = 0 , 𝑇 𝑥 ( 𝑡 ) = 𝜙 ( 𝑡 ) , 𝑡 ∈ − ∞ , 0 ( 1 . 1 ) where 𝑇 > 0 , 0 < 𝛼 < 1 , − 𝐴 is a generator of an analytic semigroup ( 𝑇 ( 𝑡 ) ) 𝑡 ≥ 0 on a Banach space 𝕏 such that ‖ 𝑇 ( 𝑡 ) ‖ ≤ 𝐾 for all 𝑡 ≥ 0 and ‖ 𝐴 𝑇 ( 𝑡 ) 𝑥 ‖ ≤ 𝐾 / 𝑡 ‖ 𝑥 ‖ for every 𝑥 ∈ 𝑋 and 𝑡 > 0 . The function 𝑓 ∶ 𝐼 × ℬ → 𝕏 is continuous functions with additional assumptions.

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

𝐷 𝛼 1 ℎ ( 𝑡 ) = Γ ( 1 − 𝛼 ) 𝑡 0 ( 𝑡 − 𝑠 ) − 𝛼 ℎ ′ ( 𝑠 ) 𝑑 𝑠 . ( 1 . 2 ) 𝜙 ∈ ℬ where ℬ is called phase space to be defined in Section 2. For any function 𝑥 defined on ] − ∞ , 𝑇 ] and any 𝑡 ∈ 𝐼 , we denote by 𝑥 𝑡 the element of ℬ defined by

𝑥 𝑡 ] ] . ( 𝜃 ) = 𝑥 ( 𝑡 + 𝜃 ) , 𝜃 ∈ − ∞ , 0 ( 1 . 3 ) The function 𝑥 𝑡 represents the history of the state from − ∞ up to the present time 𝑡 .

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

𝐷 𝛼 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝑡 𝑛 [ ] 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝐵 𝑥 ( 𝑡 ) ) , 𝑡 ∈ 0 , 𝑇 , 𝑛 ∈ ℤ + , 𝑥 ( 0 ) = 𝑥 0 + 𝑔 ( 𝑥 ) , ( 1 . 4 ) where 𝑇 is a positive real, 0 < 𝛼 < 1 , 𝐴 is the generator of a 𝐶 0 -semigroup ( 𝑆 ( 𝑡 ) ) 𝑡 ≥ 0 on a Banach space 𝕏 , ∫ 𝐵 𝑥 ( 𝑡 ) ∶ = 𝑡 0 𝐾 ( 𝑡 , 𝑠 ) 𝑥 ( 𝑠 ) 𝑑 𝑠 , 𝐾 ∈ 𝐶 ( 𝐷 , ℝ + ) with 𝐷 defined as above and

𝐵 ∗ = s u p 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 0 𝐾 ( 𝑡 , 𝑠 ) 𝑑 𝑠 < ∞ , ( 1 . 5 ) 𝑓 ∶ ℝ × 𝕏 × 𝕏 → 𝑋 is a nonlinear function, 𝑔 ∶ 𝒞 ( [ 0 , 𝑇 ] , 𝕏 ) → 𝐷 ( 𝐴 ) is continuous, and 0 < 𝑞 < 1 . The derivative 𝐷 𝛼 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 ℬ 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

𝐷 𝛼 𝑥 ( 𝑡 ) = 𝑓 𝑡 , 𝑥 𝑡 [ ] ] ] , , 𝑡 ∈ 𝐼 = 0 , 𝑇 , 0 < 𝛼 < 1 , 𝑥 ( 𝑡 ) = 𝜙 ( 𝑡 ) , 𝑡 ∈ − ∞ , 0 ( 1 . 6 ) where 𝐷 𝛼 is the standard Riemann-Liouville fractional derivative, 𝜙 in the phase space ℬ , with 𝜙 ( 0 ) = 0 .

2. Preliminaries

From now on, we set 𝐼 = [ 0 , 𝑇 ] . We denote by 𝕏 a Banach space with norm ‖ ⋅ ‖ , 𝒞 ( 𝐼 ; 𝕏 ) the space of all 𝕏 -valued continuous functions on 𝐼 , and 𝐿 ( 𝕏 ) the Banach space of all linear and bounded operators on 𝕏 .

We assume that the phase space ( ℬ , ‖ ⋅ ‖ ℬ ) is a seminormed linear space of functions mapping ] − ∞ , 0 ] into 𝕏 , and satisfying the following fundamental axioms due to Hale and Kato (see e.g., in [21]).

( 𝐴 0 ) If 𝑥 ∶ ] − ∞ , 𝑇 ] → 𝕏 , is continuous on 𝐼 and 𝑥 0 ∈ ℬ , then for every 𝑡 ∈ 𝐼 the following conditions hold: (i) 𝑥 𝑡 is in ℬ , (ii) ‖ 𝑥 ( 𝑡 ) ‖ ≤ 𝐻 ‖ 𝑥 𝑡 ‖ ℬ , (iii) ‖ 𝑥 𝑡 ‖ ℬ ≤ 𝐶 1 ( 𝑡 ) s u p 0 ≤ 𝑠 ≤ 𝑡 ‖ 𝑥 ( 𝑠 ) ‖ + 𝐶 2 ( 𝑡 ) ‖ 𝑥 0 ‖ ℬ ,

where 𝐻 ≥ 0 is a constant, 𝐶 1 ∶ [ 0 , + ∞ [ → [ 0 , + ∞ [ is continuous, 𝐶 2 ∶ [ 0 , + ∞ [ → [ 0 , + ∞ [ is locally bounded, and 𝐻 , 𝐶 1 , 𝐶 2 are independent of 𝑥 ( ⋅ ) .

( 𝐴 1 ) For the function 𝑥 ( ⋅ ) in ( 𝐴 0 ) , 𝑥 𝑡 is a ℬ -valued continuous function on 𝐼 . ( 𝐴 2 ) The space ℬ is complete.

Remark. Condition (ii) in ( 𝐴 0 ) is equivalent to ‖ 𝜙 ( 0 ) ‖ ≤ 𝐻 ‖ 𝜙 ‖ ℬ , for all 𝜙 ∈ ℬ .

Let us recall some examples of phase spaces.

Example. (E1) 𝐵 𝑈 𝐶 ( ] − ∞ , 0 ] ) ; 𝕏 ) the Banach space of all bounded and uniformly continuous functions 𝜙 ∶ ] − ∞ , 0 ] → 𝕏 endowed with the supnorm.
(E2) 𝐶 0 ( ] − ∞ , 0 ] ) ∶ 𝕏 ) the Banach space of all bounded and continuous functions 𝜙 ∶ ] − ∞ , 0 ] → 𝕏 such that l i m 𝜃 → − ∞ 𝜙 ( 𝜃 ) = 0 endowed with the norm | | 𝜙 | | ∶ = s u p 𝜃 ≤ 0 | | | | 𝜙 ( 𝜃 ) . ( 2 . 1 )
( E 3 ) 𝐶 𝛾 ∶ = { 𝜙 ∈ 𝐶 ( ] − ∞ , 0 ] ∶ 𝕏 ) ∶ l i m 𝜃 → − ∞ 𝑒 𝛾 𝜃 𝜙 ( 𝜃 ) e x i s t s i n 𝕏 } endowed with the norm | | 𝜙 | | = s u p − ∞ < 𝜃 ≤ 0 𝑒 𝛾 𝜃 | | | | . 𝜙 ( 𝜃 ) ( 2 . 2 )
Note that the space 𝐶 𝛾 is a uniform fading memory for 𝛾 > 0 .

Throughout this work 𝑓 will be a continuous function 𝐼 × ℬ × → 𝕏 . Let Ω be set defined by:

] ] Ω = 𝑥 ∶ − ∞ , 𝑇 ⟶ 𝕏 s u c h t h a t 𝑥 ∣ ] − ∞ , 0 ] ∈ ℬ , 𝑥 ∣ 𝐼 ∈ 𝒞 ( 𝐼 ; 𝕏 ) . ( 2 . 3 )

Remark. We recall that the Cauchy Problem 𝐷 𝛼 [ ] , 𝑥 ( 𝑡 ) + 𝐴 𝑥 ( 𝑡 ) = 0 , 𝑡 ∈ 0 , 𝑇 𝑥 ( 0 ) = 𝑥 0 ∈ 𝐷 , 𝑡 ∈ 𝐼 , ( 2 . 4 ) where − 𝐴 is a closed linear operator defined on a dense subset, 𝐷 ⊂ 𝑋 is wellposed, and the unique solution is given by 𝑥 ( 𝑡 ) = ∞ 0 𝜁 𝛼 ( 𝜎 ) 𝑇 ( 𝑡 𝛼 𝜎 ) 𝑥 0 𝑑 𝜎 , ( 2 . 5 ) where 𝜁 𝛼 is a probability density function defined on ( 0 , ∞ ) such that its Laplace transform is given by ∞ 0 𝑒 − 𝜎 𝑥 𝜁 𝛼 ( 𝜎 ) 𝑑 𝜎 = ∞ 𝑖 = 0 − 𝑥 𝑖 Γ ( 1 + 𝛼 𝑖 ) , 𝑥 > 0 ( 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 𝑥 ∈ Ω is said to be a mild solution of (1.1) if 𝑥 satisfies ⎧ ⎪ ⎨ ⎪ ⎩ ] ] , 𝑄 𝑥 ( 𝑡 ) = 𝜙 ( 𝑡 ) , 𝑡 ∈ − ∞ , 0 ( 𝑡 ) 𝜙 ( 0 ) + 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑥 𝑠 𝑑 𝑠 , 𝑡 ∈ 𝐼 , ( 2 . 7 ) where 𝑄 ( 𝑡 ) = ∞ 0 𝜁 𝛼 ( 𝜎 ) 𝑇 ( 𝑡 𝛼 𝜎 ) 𝑑 𝜎 , 𝑅 ( 𝑡 ) = 𝛼 ∞ 0 𝜎 𝑡 𝛼 − 1 𝜁 𝛼 ( 𝜎 ) 𝑇 ( 𝑡 𝛼 𝜎 ) 𝑑 𝜎 . ( 2 . 8 )

Remark 2.5. Note that ‖ 𝑅 ( 𝑡 ) ‖ 𝐵 ( 𝕏 ) ≤ 𝛼 𝐾 𝑡 𝛼 − 1 , 𝑡 ≥ 0 , ( 2 . 9 ) since ∫ ∞ 0 𝜎 𝜁 𝛼 ( 𝜎 ) 𝑑 𝜎 = 1 (cf. [23]).

3. Main Results

We present now our result.

Theorem 3.1. Assume the following. ( 𝐻 1 ) There exist 𝜇 > 0 such that for all 𝑡 ∈ 𝐼 , ( 𝜓 , 𝜑 ) ∈ ℬ 2 ‖ 𝑓 ( 𝑡 , 𝜑 ) − 𝑓 ( 𝑡 , 𝜓 ) ‖ ≤ 𝜇 ‖ 𝜑 − 𝜓 ‖ ℬ ( 3 . 1 ) ( 𝐻 2 ) There exists 𝛿 , with 0 < 𝛿 < 1 such that the function Λ ∶ 𝐼 → ] 0 , + ∞ ] defined by: Λ ( 𝑡 ) = 𝜇 𝐾 𝐶 ∗ 1 𝑡 𝛼 ( 3 . 2 ) satisfies Λ ( 𝑡 ) ≤ 𝛿 for all 𝑡 ∈ 𝐼 . Here 𝐶 ∗ 1 = s u p 𝑡 ∈ 𝐼 𝐶 1 ( 𝑡 ) . ( 3 . 3 ) Then (1.1) has a unique mild solution on ] − ∞ , 𝑇 ] .

Proof. Consider the operator 𝑁 ∶ Ω → Ω defined by ⎧ ⎪ ⎨ ⎪ ⎩ ] ] , 𝑄 𝑁 ( 𝑥 ) ( 𝑡 ) = 𝜙 ( 𝑡 ) , 𝑡 ∈ − ∞ , 0 ( 𝑡 ) 𝜙 ( 0 ) + 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑥 𝑠 𝑑 𝑠 , 𝑡 ∈ 𝐼 . ( 3 . 4 ) Let 𝑦 ( ⋅ ) ∶ ] − ∞ , 𝑇 ] → 𝕏 be the function defined by ] ] , 𝑦 ( 𝑡 ) = 𝜙 ( 𝑡 ) , i f 𝑡 ∈ − ∞ , 0 𝑄 ( 𝑡 ) 𝜙 ( 0 ) , i f 𝑡 ∈ 𝐼 . ( 3 . 5 ) Then 𝑦 0 = 𝜙 . For each 𝑧 ∈ 𝒞 ( 𝐼 , 𝕏 ) with 𝑧 ( 0 ) = 0 , we denote by 𝑧 the function defined by ] ] , 𝑧 ( 𝑡 ) = 0 , 𝑡 ∈ − ∞ , 0 𝑧 ( 𝑡 ) , 𝑡 ∈ 𝐼 . ( 3 . 6 )
If 𝑥 ( ⋅ ) verifies (2.7) then writing 𝑥 ( 𝑡 ) = 𝑦 ( 𝑡 ) + 𝑧 ( 𝑡 ) for 𝑡 ∈ 𝐼 , we have 𝑥 𝑡 = 𝑦 𝑡 + 𝑧 𝑡 for 𝑡 ∈ 𝐼 and
∫ 𝑧 ( 𝑡 ) = 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑦 𝑠 + 𝑧 𝑠 𝑑 𝑠 . ( 3 . 7 )
Moreover 𝑧 0 = 0 .
Let
𝑍 0 = 𝑧 ∈ Ω s u c h t h a t 𝑧 0 = 0 . ( 3 . 8 ) For any 𝑧 ∈ 𝑍 0 , we have ‖ 𝑧 ‖ 𝑍 0 = s u p 𝑡 ∈ 𝐼 ‖ ‖ 𝑧 ‖ 𝑧 ( 𝑡 ) ‖ + 0 ‖ ‖ ℬ = s u p 𝑡 ∈ 𝐼 ‖ 𝑧 ( 𝑡 ) ‖ . ( 3 . 9 ) Thus ( 𝑍 0 , ‖ ⋅ ‖ 𝑍 0 ) is a Banach space. We define the operator Π ∶ 𝑍 0 → 𝑍 0 by ∫ Π ( 𝑧 ) ( 𝑡 ) = 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑦 𝑠 + 𝑧 𝑠 𝑑 𝑠 . ( 3 . 1 0 ) It is clear that the operator 𝑁 has a unique fixed point if and only if Π has a unique fixed point. So let us prove that Π has a unique fixed point. Observe first that Π is obviously well defined. Now, consider 𝑧 , 𝑧 ∗ ∈ 𝑍 0 . For any 𝑡 ∈ 𝐼 , we have ‖ ‖ 𝑧 Π ( 𝑧 ) ( 𝑡 ) − Π ∗ ‖ ‖ = | | | | ( 𝑡 ) 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑦 𝑠 + 𝑧 𝑠 𝑑 𝑠 − 𝑡 0 𝑅 ( 𝑡 − 𝑠 ) 𝑓 𝑠 , 𝑦 𝑠 + 𝑧 ∗ 𝑠 | | | | ≤ 𝑑 𝑠 𝑡 0 ‖ ‖ ‖ 𝑓 ‖ 𝑅 ( 𝑡 − 𝑠 ) 𝑠 , 𝑦 𝑠 + 𝑧 𝑠 − 𝑓 𝑠 , 𝑦 𝑠 + 𝑧 ∗ 𝑠 ‖ ‖ 𝑑 𝑠 ≤ 𝜇 𝑡 0 ‖ ‖ ‖ 𝑅 ( 𝑡 − 𝑠 ) ‖ 𝑧 𝑠 − 𝑧 ∗ 𝑠 ‖ ‖ ℬ 𝑑 𝑠 . ( 3 . 1 1 )
So using ( 𝐴 0 ) - ( i i i ) , (2.9) and (3.3), we obtain for all 𝑡 ∈ 𝐼
‖ ‖ 𝑧 Π ( 𝑧 ) ( 𝑡 ) − Π ∗ ( ‖ ‖ 𝑡 ) ≤ 𝜇 𝛼 𝐾 𝑡 0 ( 𝑡 − 𝑠 ) 𝛼 − 1 𝐶 1 ( 𝑠 ) ‖ 𝑧 − 𝑧 ∗ ‖ 𝑍 0 ≤ 𝜇 𝐾 𝐶 ∗ 1 𝑡 𝛼 ‖ 𝑧 − 𝑧 ∗ ‖ 𝑍 0 ( 3 . 1 2 ) which according to ( 𝐻 2 ) gives ‖ ‖ Π 𝑧 ( 𝑧 ) ( 𝑡 ) − Π ∗ ‖ ‖ ( 𝑡 ) ≤ Λ ( 𝑡 ) ‖ 𝑧 − 𝑧 ∗ ‖ 𝑍 0 ≤ 𝛿 ‖ 𝑧 − 𝑧 ∗ ‖ 𝑍 0 . ( 3 . 1 3 )
Therefore
‖ Π 𝑧 − Π 𝑧 ∗ ‖ 𝑍 0 ≤ 𝛿 ‖ 𝑧 − 𝑧 ∗ ‖ 𝑍 0 . ( 3 . 1 4 )
And since 0 ≤ 𝛿 < 1 , we conclude by way of the Banach's contraction mapping principle that Π has a unique fixed point 𝑧 ∈ 𝑍 0 . This means that 𝑁 has a unique fixed point 𝑥 ∈ Ω which is obviously a mild solution of (1.1) on ] − ∞ , 𝑇 ] .

4. Application

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

𝐷 𝛼 𝑡 𝜕 𝑢 ( 𝑡 , 𝜉 ) = 2 𝜕 𝜉 2 𝑢 ( 𝑡 , 𝜉 ) + 0 − ∞ 𝜂 ( 𝜎 ) 𝑢 ( 𝑡 + 𝜎 , 𝜉 ) 𝑑 𝜎 , 0 ≤ 𝜉 ≤ 𝜋 , 𝑢 ( 𝑡 , 0 ) = 𝑢 ( 𝑡 , 𝜋 ) = 0 f o r 𝑡 ∈ ℝ , ( 4 . 1 ) where 0 < 𝛼 < 1 and 𝜂 is a positive function on ( − ∞ , 0 ] with ∫ 0 − ∞ 𝜂 ( 𝜎 ) 𝑑 𝜎 < ∞ .

Now let 𝑋 = 𝐿 2 ( 0 , 𝜋 ) and consider the operator 𝐴 ∶ 𝐷 ( 𝐴 ) ⊂ 𝑋 → 𝑋 defined by

𝐷 ( 𝐴 ) = 𝐻 2 ( 0 , 𝜋 ) ∩ 𝐻 1 0 𝐻 ( 0 , 𝜋 ) = 2 , ( 0 , 𝜋 ) ∶ 𝑧 ( 0 ) = 𝑧 ( 𝜋 ) = 0 𝐴 𝑧 = 𝑧 . ( 4 . 2 ) Clearly 𝐷 ( 𝐴 ) is dense in 𝐿 2 ( 0 , 𝜋 ) .

Define

𝑓 ( 𝜙 ) ( 𝜉 ) ∶ = ∞ 0 [ ] 𝜂 ( 𝜎 ) 𝜙 ( 𝜎 ) ( 𝜉 ) 𝑑 𝜎 , 𝜉 ∈ 0 , 𝜋 , 𝜙 ∈ ℬ . ( 4 . 3 ) We choose ℬ as in Example (E3) above. Put

] [ ] . 𝑥 ( 𝑡 ) ( 𝜉 ) = 𝑢 ( 𝑡 , 𝜉 ) , 𝑡 ∈ ( − ∞ , 𝑇 , 𝜉 ∈ 0 , 𝜋 ( 4 . 4 ) Then we get

𝐷 𝛼 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝑓 𝑡 , 𝑥 𝑡 , ( 4 . 5 ) where 𝑓 ( 𝑡 , 𝑥 ) is obviously Lipschitzian in 𝑥 uniformly in 𝑡 . Thus we can state what follows.

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

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