- Open Access
Global attractivity for fractional order delay partial integro-differential equations
© Abbas et al; licensee Springer. 2012
- Received: 22 March 2012
- Accepted: 15 May 2012
- Published: 15 May 2012
Our aim in this work is to study the existence and the attractivity of solutions for a system of delay partial integro-differential equations of fractional order. We use the Schauder fixed point theorem for the existence of solutions, and we prove that all solutions are locally asymptotically stable.
AMS (MOS) Subject Classifications: 26A33.
- delay integro-differential equation
- left-sided mixed Riemann-Liouville integral of fractional order
- Caputo fractional-order derivative
- fixed point
Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as the differential calculus and it has been developed up to nowadays (see Kilbas et al. , Hilfer ). Fractional differential and integral equations have recently been applied in various areas of Engineering, Mathematics, Physics and Bio-engineering and so on. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Baleanu et al. , Hilfer , Kilbas et al. , Lakshmikantham et al. , Podlubny , and the articles by Abbas et al. [6–8], Vityuk and Golushkov . Recently interesting results of the stability of the solutions of various classes of integral equations of fractional order have obtained by Banaś et al. [10, 11], Darwish et al. , Dhage [13, 14] and the references therein.
Where b > 0, θ = (0, 0) ℝ+ = [0, ∞), τ i , ξ i ≥ 0; i = 1..., m, , , is the Caputo fractional derivative of order r = (r1,r2) ∈ (0, ∞)×(0, ∞), is the partial Riemann-Liouville integral of order r2 with respect to x, f : J × ℝ m → ℝ is a given continuous function, φ : ℝ+ → ℝ, ψ : [0, b] → ℝ are absolutely continuous functions with limt→∞φ(t) = 0, and ψ(x) = φ(0) for each x ∈ [0, b], and is continuous with φ(t) = Φ(t, 0) for each t ∈ ℝ+, and ψ(x) = Φ(0, x) for each x ∈ [0, b].
This article initiates the question of local attractivity of the solution of problem (1)-(3).
For u0 ∈ BC and η ∈ (0, ∞), we denote by B(u0, η), the closed ball in BC centered at u0 with radius η.
where σ = (1, 1). For instance, exists for all r1, r2 ∈ (0, ∞), when u ∈ L1([0, a] × [0, b]).
By 1 − r we mean (1 − r1, 1 − r2) ∈ (0, 1] × (0, 1]. Denote by , the mixed second order partial derivative.
Definition 2.4Let r ∈ (0, 1] × (0, 1] and u ∈ L1([0, a] × [0, b]). The Caputo fractional-order derivative of order r of u is defined by the expression.
When the limit (5) is uniform with respect to B(u0, η) ∩ Ω, solutions of Equation (4) are said to be uniformly locally attractive (or equivalently that solutions of (4) are asymptotically stable).
Lemma 2.7Let D ⊂ BC. Then D is relatively compact in BC if the following conditions hold:
(a) D is uniformly bounded in BC,
(b) The functions belonging to D are almost equicontinuous on ℝ+ × [0, b], i.e., equicontinuous on every compact of ℝ+ × [0, b],
(c) The functions from D are equiconvergent, that is, given ε > 0, x ∈ [0, b] there corresponds T (ε, x) > 0 such that |u(t, x)-limt→∞u(t, x)| < ε for any t ≥ T (ε, x) and u ∈ D.
Let us start by defining what we mean by a solution of the problem (1)-(3).
Definition 3.1 A function u ∈ BC is said to be a solution of (1)-(3) if u satisfies Equation
(1) on J, Equation (2) onand condition (3) is satisfied.
Now, we shall prove the following theorem concerning the existence and the attractivity of a solution of problem (1)-(3).
Theorem 3.3 Assume that the function f satisfying the following hypothesis
Then the problem (1)-(3) has at least one solution in the space BC. Moreover, solutions of problem (1)-(3) are uniformly locally attractive.
Hence, N(u) ∈ BC. This proves that the operator N maps BC into itself.
By Lemma 3.2, the problem of finding the solutions of the problem (1)-(3) is reduced to finding the solutions of the operator equation N(u)= u. Equation (7) yields that N transforms the ball B η := B(0, η) into itself. We shall show that N : B η → B η satisfies the assumptions of Schauder's fixed point theorem . The proof will be given in several steps and cases.
Step 1: N is continuous.
Step 2: N(B η ) is uniformly bounded.
This is clear since N(B η ) ⊂ B η and B η is bounded.
Step 3: N(B η ) is equicontinuous on every compact subset [−T, a] × [−ξ, b] of [−T, a] × [−ξ, ∞), a > 0.
From continuity of φ, p i ; i = 0,...,m and as t1 → t2 and x1 → x2, the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t1 < t2 < 0, x1 < x2 < 0 and t1 ≤ 0 ≤ t2, x1 ≤ 0 ≤ x2 is obvious.
Step 4: N(B η ) is equiconvergent.
As a consequence of Steps 1-4 together with the Lemma 2.7, we can conclude that N : B η → B η is continuous and compact. From an application of Schauder's theorem , we deduce that N has a fixed point u which is a solution of the problem (1)-(3).
Consequently, all solutions of problem (1)-(3) are uniformly locally attractive.
We have for each x ∈ [0, ∞), μ(t) = e-t→ 0 as t → ∞.
Hence by Theorem 3.3, the problem (11)-(13) has a solution defined on and all solutions are uniformly locally attractive on
The authors were grateful to the anonymous referees for their valuable comments and remarks which were taken into account in the revised version of the manuscript.
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V, Amsterdam; 2006.Google Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, New Jersey; 2000.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific Publishing, New York; 2012.Google Scholar
- Lakshmikantham V, Leela S, Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Abbas S, Benchohra M: Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Anal Hybrid Syst 2009, 3: 597–604. 10.1016/j.nahs.2009.05.001MathSciNetView ArticleGoogle Scholar
- Abbas S, Benchohra M: Darboux problem for implicit impulsive partial hyperbolic differential equations. Electron J Diff Equ 2011, 2011: 15.MathSciNetView ArticleGoogle Scholar
- Abbas S, Benchohra M, Vityuk AN: On fractional order derivatives and Darboux problem for implicit differential equations. Fract Calc Appl Anal 2012, 15: 168–182.MathSciNetView ArticleGoogle Scholar
- Vityuk AN, Golushkov AV: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil 2004, 7: 318–325. 10.1007/s11072-005-0015-9MathSciNetView ArticleGoogle Scholar
- Banaś J, Dhage BC: Global asymptotic stability of solutions of a functional integral equation. Nonlinear Anal Theory Methods Appl 2008, 69: 1945–1952. 10.1016/j.na.2007.07.038View ArticleGoogle Scholar
- Banaś J, Zając T: A new approach to the theory of functional integral equations of fractional order. J Math Anal Appl 2011, 375: 375–387. 10.1016/j.jmaa.2010.09.004MathSciNetView ArticleGoogle Scholar
- Darwish MA, Henderson J, O'Regan D: Existence and asymptotic stability of solutions of a perturbed fractional functional integral equations with linear modification of the argument. Bull Korean Math Soc 2011, 48: 539–553. 10.4134/BKMS.2011.48.3.539MathSciNetView ArticleGoogle Scholar
- Dhage BC: Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem. Nonlinear Anal 2009, 70: 2485–2493. 10.1016/j.na.2008.03.033MathSciNetView ArticleGoogle Scholar
- Dhage BC: Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness. Diff Equ Appl 2010, 2: 299–318.MathSciNetGoogle Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory Appl Gordon Breach Yverdon 1993.Google Scholar
- Corduneanu C: Integral Equations and Stability of Feedback Systems. Acedemic Press, New York; 1973.Google Scholar
- Granas A, Dugundji J: Fixed Point Theory. Springer-Verlag New York; 2003.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.