Theory and Modern Applications

Global attractivity for fractional order delay partial integro-differential equations

Abstract

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.

1. Introduction

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. [1], Hilfer [2]). 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. [3], Hilfer [2], Kilbas et al. [1], Lakshmikantham et al. [4], Podlubny [5], and the articles by Abbas et al. [68], Vityuk and Golushkov [9]. 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. [12], Dhage [13, 14] and the references therein.

In this article, we established sufficient conditions for the existence and the attractivity of solutions of the following system of delay integro-differential equations of fractional order of the form

$c{D}_{\theta }^{r}u\left(t,x\right)=f\left(t,x,{I}_{0,x}^{{r}_{2}}u\left(t,x\right),u\left(t-{\tau }_{1},x-{\xi }_{1}\right),\dots ,u\left(t-{\tau }_{m},x-{\xi }_{m}\right)\right);$
(1)
(2)
$\left\{\begin{array}{cc}\hfill u\left(t,\phantom{\rule{2.77695pt}{0ex}}0\right)=\phi \left(t\right);\hfill & \hfill t\in \left[0,\phantom{\rule{2.77695pt}{0ex}}\infty \right),\hfill \\ \hfill u\left(0,\phantom{\rule{2.77695pt}{0ex}}x\right)=\psi \left(x\right);\hfill & \hfill x\in \left[0,\phantom{\rule{2.77695pt}{0ex}}b\right],\hfill \end{array}\right\\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}$
(3)

Where b > 0, θ = (0, 0) + = [0, ∞), τ i , ξ i ≥ 0; i = 1..., m, $T=\underset{i=1,\dots m}{\text{max}}\left\{{\tau }_{i}\right\}$, $\xi =\underset{i=1,\dots m}{\text{max}}\left\{{\xi }_{i}\right\}$, $c{D}_{\theta }^{r}$ is the Caputo fractional derivative of order r = (r1,r2) (0, ∞)×(0, ∞), ${I}_{0,x}^{{r}_{2}}$ is the partial Riemann-Liouville integral of order r2 with respect to x, f : J × m is a given continuous function, φ : +, ψ : [0, b] → are absolutely continuous functions with limt→∞φ(t) = 0, and ψ(x) = φ(0) for each x [0, b], and $\Phi :\stackrel{̃}{J}\to {ℝ}^{n}$ is continuous with φ(t) = Φ(t, 0) for each t +, 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).

2. Preliminaries

In the following, we present briefly notations, definitions, and preliminary facts which are used throughout this article. By L1([0, a] × [0, b]); a, b > 0, we denote the space of Lebesgue-integrable functions u : [0, a] × [0, b] → with the norm

$||u||1=\underset{0}{\overset{a}{\int }}\underset{0}{\overset{b}{\int }}|u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|dxdt.$

By BC := BC([−T, ∞)×[−ξ, b]) we denote the Banach space of all bounded and continuous functions from [−T, ∞) × [−ξ, b] into equipped with the standard norm

$||u||BC=\underset{\left(t,x\right)\in \left[-T,\infty \right)×\left[-\xi ,b\right]}{\text{sup}}|u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|.$

For u0 BC and η (0, ∞), we denote by B(u0, η), the closed ball in BC centered at u0 with radius η.

Definition 2.1[15]Let ρ (0, ∞) and u L1([0, a]×[0, b]),a, b > 0. The partial Riemann-Liouville integral of order ρ of u(t, x) with respect to t is defined by the expression

${I}_{0,t}^{r}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\frac{1}{\Gamma \left(\rho \right)}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\rho -1}u\left(s,\phantom{\rule{2.77695pt}{0ex}}x\right)ds,\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}almost\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right].$

Analogously, we define the integral

Definition 2.2[9]Let r = (r1, r2) (0, ∞) × (0, ∞), θ = (0, 0) and u L1([0, a] × [0, b]). The left-sided mixed Riemann-Liouville integral of order r of u is defined by

$\left({I}_{\theta }^{r}u\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}u\left(s,\phantom{\rule{2.77695pt}{0ex}}\tau \right)dsd\tau .$

In particular,

$\left({I}_{\theta }^{\theta }u\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right),\left({I}_{\theta }^{\sigma }u\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau ;\phantom{\rule{0.3em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{almost}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{all}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right],$

where σ = (1, 1). For instance, ${I}_{\theta }^{r}u$ exists for all r1, r2 (0, ∞), when u L1([0, a] × [0, b]).

Note also that when u C([0, a] × [0, b]), then $\left({I}_{\theta }^{r}u\right)\in C\left(\left[0,a\right]×\left[0,b\right]\right)$, moreover

$\left({I}_{\theta }^{r}u\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}0\right)=\left({I}_{\theta }^{r}u\right)\left(0,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;t\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right],\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\phantom{\rule{2.77695pt}{0ex}}b\right].$

Example 2.3 Let λ, ω (−1, ∞) and r = (r1, r2) (0, ∞) × (0, ∞), then

${I}_{\theta }^{r}{t}^{\lambda }{x}^{\omega }=\frac{\Gamma \left(1+\lambda \right)\Gamma \left(1+\omega \right)}{\Gamma \left(1+\lambda +{r}_{1}\right)\Gamma \left(1+\omega +{r}_{2}\right)}{t}^{\lambda +{r}_{1}}{x}^{\omega +{r}_{2}},for\phantom{\rule{0.3em}{0ex}}almost\phantom{\rule{0.3em}{0ex}}all\phantom{\rule{0.3em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right].$

By 1 − r we mean (1 − r1, 1 − r2) (0, 1] × (0, 1]. Denote by ${D}_{tx}^{2}:=\frac{{\partial }^{2}}{\partial t\partial x}$, the mixed second order partial derivative.

Definition 2.4[9]Let 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$c{D}_{\theta }^{r}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\left({I}_{\theta }^{1-r}{D}_{tx}^{2}u\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)$.

The case σ = (1, 1) is included and we have

Example 2.5 Let λ, ω (−1, ∞) and r = (r1, r2) (0, 1] × (0, 1], then

$c{D}_{\theta }^{r}{t}^{\lambda }{x}^{\omega }=\frac{\Gamma \left(1+\lambda \right)\Gamma \left(1+\omega \right)}{\Gamma \left(1+\lambda -{r}_{1}\right)\Gamma \left(1+\omega -{r}_{2}\right)}{t}^{\lambda -{r}_{1}}{x}^{\omega -{r}_{2}},\phantom{\rule{0.3em}{0ex}}for\phantom{\rule{0.3em}{0ex}}almost\phantom{\rule{0.3em}{0ex}}all\phantom{\rule{0.3em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right].$

Let ≠ Ω BC, and let G : Ω → Ω, and consider the solutions of equation

$\mathsf{\text{(}}Gu\mathsf{\text{)(}}t,x\mathsf{\text{)}}=u\mathsf{\text{(}}t,x\mathsf{\text{)}}.$
(4)

Inspired by the definition of the attractivity of solutions of integral equations (see for instance [10]), we introduce the following concept of attractivity of solutions for Equation (4).

Definition 2.6 Solutions of Equation (4) are locally attractive if there exists a ball B(u0, η) in the space BC such that, for arbitrary solutions v = v(t, x) and w = w(t, x) of Equation (4) belonging to B(u0, η) ∩ Ω, we have that, for each × [0, b],

$\underset{t\to \infty }{\text{lim}}\left(v\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)-w\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\right)=0.$
(5)

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 2.7[16]Let 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 + × [0, b], i.e., equicontinuous on every compact of + × [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 tT (ε, x) and u D.

3. Main results

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) on$\stackrel{̃}{J}$and condition (3) is satisfied.

Lemma 3.2[6]Let f L1([0, a] × [0, b]); a, b > 0. A function u AC([0, a] × [0, b]) is a solution of problem

$\left\{\begin{array}{l}{\left(}^{c}{D}_{\theta }^{r}u\right)\left(t,x\right)=f\left(t,x\right);\left(t,x\right)\in \left[0,a\right]×\left[0,b\right],\hfill \\ u\left(t,0\right)=\phi \left(t\right);t\in \left[0,a\right],u\left(0,x\right)=\psi \left(x\right);x\in \left[0,b\right],\hfill \\ \phi \left(0\right)=\psi \left(0\right),\hfill \end{array}$

if and only if u(t, x) satisfies

$u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\mu \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)+\left({I}_{\theta }^{r}f\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right);\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left[0,\phantom{\rule{2.77695pt}{0ex}}a\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right],$

where

$\mu \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\phi \left(t\right)+\psi \left(x\right)-\phi \left(0\right).$

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

(H) There exists continuous functions p i : + × [0, b] → +such that

$\left(1+\sum _{i=0}^{m}|{u}_{i}|\right)\left|f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}{u}_{0},\phantom{\rule{2.77695pt}{0ex}}{u}_{1},\phantom{\rule{2.77695pt}{0ex}}{u}_{2},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{u}_{m}\right)\right|\le \sum _{i=0}^{m}|{u}_{i}|{p}_{i}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right);$

for (t, x) + × [0, b] and for u i ; i = 0,..., m. Moreover, assume that

$\underset{t\to \infty }{\text{lim}}{I}_{\theta }^{r}{p}_{i}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;x\in \left[0,\phantom{\rule{2.77695pt}{0ex}}b\right];i=0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}m.$

Then the problem (1)-(3) has at least one solution in the space BC. Moreover, solutions of problem (1)-(3) are uniformly locally attractive.

Proof. Set

${\Phi }^{*}:=\underset{\left(t,x\right)\in \stackrel{̃}{J}}{\text{sup}}\Phi \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right),{\phi }^{*}:=\underset{t\in {ℝ}_{+}}{\text{sup}}\phi \left(t\right)\phantom{\rule{0.3em}{0ex}}and\phantom{\rule{0.3em}{0ex}}{p}_{i}^{*}:=\underset{\left(t,x\right)\in {ℝ}_{+}×\left[0,b\right]}{\text{sup}}{I}_{\theta }^{r}{p}_{i}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right);i=0,\dots ,m.$

From (H), we infer that ${p}_{i}^{*};i=0,\dots ,m$ are finite. Let us define the operator N such that, for any u BC,

$\left(Nu\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=\left\{\begin{array}{cc}\hfill \Phi \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right);\hfill & \hfill \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \stackrel{̃}{J},\hfill \\ \hfill \phi \left(t\right)\hfill \\ \hfill +{I}_{\theta }^{r}f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,x}^{{r}_{2}}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right),\phantom{\rule{2.77695pt}{0ex}}u\left(t-{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}x-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(t-{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}x-{\xi }_{m}\right)\right);\hfill & \hfill \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in J.\hfill \end{array}\right\$
(6)

The operator N maps BC into BC; Indeed the map N(u) is continuous on [−T, ∞) × [−ξ, b] for any u BC, and for each (t, x) J we have

$\begin{array}{l}\phantom{\rule{1em}{0ex}}|\left(Nu\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|\le |\phi \left(t\right)|\phantom{\rule{2em}{0ex}}\\ +|{I}_{\theta }^{r}f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,x}^{{r}_{2}}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right),\phantom{\rule{2.77695pt}{0ex}}u\left(t-{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}x-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(t-{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}x-{\xi }_{m}\right)\right)|\phantom{\rule{2em}{0ex}}\\ \le |\phi \left(t\right)|+\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}\phantom{\rule{2em}{0ex}}\\ ×\left(|{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)|{p}_{0}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)+\sum _{i=1}^{m}|u\left(\tau -{\tau }_{i},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{i}\right)|{p}_{i}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)\right)\phantom{\rule{2em}{0ex}}\\ ×{\left(1+|{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)|+\sum _{i=1}^{m}|u\left(\tau -{\tau }_{i},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{i}\right)|\right)}^{-1}dsd\tau .\phantom{\rule{2em}{0ex}}\\ \le {\phi }^{*}+\sum _{i=0}^{m}{p}_{i}^{*},\phantom{\rule{2em}{0ex}}\end{array}$

and for $\left(t,x\right)\in \stackrel{̃}{J}$ we have

$|\left(Nu\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|=|\Phi \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|\le {\Phi }^{*}.$

Thus,

$||N\left(u\right)|{|}_{BC}\le \text{max}\left\{{\Phi }^{*},{\phi }^{*}+\sum _{i=0}^{m}\underset{i}{\overset{*}{p}}\right\}:=\eta .$
(7)

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 [17]. The proof will be given in several steps and cases.

Step 1: N is continuous.

Let {u n }nbe a sequence such that u n u in B η . Then, for each (t, x) [−T, ∞) ×[−ξ, b], we have

(8)

Case 1. If $\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \stackrel{̃}{J}\cup \left(\left[0,\phantom{\rule{2.77695pt}{0ex}}{T}_{0}\right]×\left[0,\phantom{\rule{2.77695pt}{0ex}}b\right]\right),{T}_{0}>0$, then, since u n u as n →∞ and f, ${I}_{0,\xi }^{{r}_{2}}$, are continuous, (8) gives

$||N\left({u}_{n}\right)-N\left(u\right)|{|}_{BC}\to 0\phantom{\rule{0.3em}{0ex}}\mathsf{\text{as}}\phantom{\rule{0.3em}{0ex}}n\to \infty .$

Case 2. If (t, x) (T0, ∞) × [0, b], T0 > 0, then from (H) and (8), we get

(9)

Since u n u as n →∞ and t →∞, then (9) gives

$||N\left({u}_{n}\right)-N\left(u\right)|{|}_{BC}\to 0\phantom{\rule{0.3em}{0ex}}\mathsf{\text{as}}\phantom{\rule{0.3em}{0ex}}n\to \infty .$

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.

Let (t1, x1), (t2, x2) [0, a] × [0, b], t1 < t2, x1 < x2 and let u B η . Thus we have

$\begin{array}{l}\phantom{\rule{1em}{0ex}}|\left(Nu\right)\left({t}_{2},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right)-\left(Nu\right)\left({t}_{1},\phantom{\rule{2.77695pt}{0ex}}{x}_{1}\right)|\le |\phi \left({t}_{2}\right)-\phi \left({t}_{1}\right)|\phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{{t}_{1}}{\int }}\underset{0}{\overset{{x}_{1}}{\int }}\left[{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}-{\left({t}_{1}-\tau \right)}^{{r}_{1}-1}{\left({x}_{1}-s\right)}^{{r}_{2}-1}\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×|f\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right),\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{m}\right)\right)|dsd\tau \phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\underset{{x}_{1}}{\overset{{x}_{2}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×|f\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right),\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{m}\right)\right)|dsd\tau \phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{{t}_{1}}{\int }}\underset{{x}_{1}}{\overset{{x}_{2}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×|f\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right),\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{m}\right)\right)|dsd\tau \phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\underset{0}{\overset{{x}_{1}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×|f\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,s}^{{r}_{2}}u\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right),\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{1},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{1}\right),\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}u\left(\tau -{\tau }_{m},\phantom{\rule{2.77695pt}{0ex}}s-{\xi }_{m}\right)\right)|dsd\tau .\phantom{\rule{2em}{0ex}}\end{array}$

Thus

$\begin{array}{l}\phantom{\rule{1em}{0ex}}|\left(Nu\right)\left({t}_{2},\phantom{\rule{2.77695pt}{0ex}}{x}_{2}\right)-\left(Nu\right)\left({t}_{1},{x}_{1}\right)|\le |\phi \left({t}_{2}\right)-\phi \left({t}_{1}\right)|\phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{{t}_{1}}{\int }}\underset{0}{\overset{{x}_{1}}{\int }}\left[{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}-{\left({t}_{1}-\tau \right)}^{{r}_{1}-1}{\left({x}_{1}-s\right)}^{{r}_{2}-1}\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}×\sum _{i=0}^{m}{p}_{i}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\underset{1}{\overset{{x}_{2}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\sum _{i=0}^{m}{p}_{i}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau \phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{{t}_{1}}{\overset{{t}_{1}}{\int }}\underset{{x}_{1}}{\overset{{x}_{2}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\sum _{i=0}^{m}{p}_{i}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau \phantom{\rule{2em}{0ex}}\\ +\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\underset{0}{\overset{{x}_{1}}{\int }}{\left({t}_{2}-\tau \right)}^{{r}_{1}-1}{\left({x}_{2}-s\right)}^{{r}_{2}-1}\sum _{i=0}^{m}{p}_{i}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau .\phantom{\rule{2em}{0ex}}\end{array}$

From continuity of φ, p i ; i = 0,...,m and as t1t2 and x1x2, 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.

Let (t, x) + × [0, b] and u B η , then we have

Thus, for each x [0, b], we get

$|\left(Nu\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)|\to 0,\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}t\to +\infty .$

Hence,

$|\left(Nu\right)\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)-\left(Nu\right)\left(+\infty ,\phantom{\rule{2.77695pt}{0ex}}x\right)|\phantom{\rule{0.3em}{0ex}}\to 0,\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}t\to +\infty .$

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 [17], we deduce that N has a fixed point u which is a solution of the problem (1)-(3).

Now we investigate the uniform local attractivity for solutions of problem (1)-(3). Let us assume that u0 is a solution of problem (1)-(3) with the conditions of this theorem. Consider the ball B(u0, η*), where

$\begin{array}{l}{\eta }^{*}:=\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{\left(t,x\right)\in J}{\mathrm{sup}}\left\{\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}\\ ×|f\left(\tau ,s,{I}_{0,s}^{{r}_{2}}u\left(\tau ,s\right),u\left(\tau -{\tau }_{1},s-{\xi }_{1}\right),\dots ,u\left(\tau -{\tau }_{m},s-{\xi }_{m}\right)\right)\\ -f\left(\tau ,s,{I}_{0,s}^{{r}_{2}}{u}_{0}\left(\tau ,s\right),{u}_{0}\left(\tau -{\tau }_{1},s-{\xi }_{1}\right),\dots ,{u}_{0}\left(\tau -{\tau }_{m},s-{\xi }_{m}\right)\right)|dsd\tau ;u\in BC\right\}.\end{array}$

Taking u B(u0, η*), we have

Thus we observe that N is a continuous function such that N(B(u0, η*)) B(u0, η*). Moreover, if u is a solution of problem (1)-(3), then

$\begin{array}{l}|u\left(t,x\right)-{u}_{0}\left(t,x\right)|=|\left(Nu\right)\left(t,x\right)-\left(N{u}_{0}\right)\left(t,x\right)|\\ \le \frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}{\int }_{0}^{t}{\int }_{0}^{x}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}\\ ×|f\left(\tau ,s,{I}_{0,s}^{{r}_{2}}u\left(\tau ,s\right),u\left(\tau -{\tau }_{1},s-{\xi }_{1}\right),\dots ,u\left(\tau -{\tau }_{m},s-{\xi }_{m}\right)\right)\\ -f\left(\tau ,s,{I}_{0,s}^{{r}_{2}}{u}_{0}\left(\tau ,s\right),{u}_{0}\left(\tau -{\tau }_{1},s-{\xi }_{1}\right),\dots ,{u}_{0}\left(\tau -{\tau }_{m},s-{\xi }_{m}\right)\right)|dsd\tau \\ \le \frac{2}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}{\int }_{0}^{t}{\int }_{0}^{x}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}\sum _{i=0}^{m}{p}_{i}\left(\tau ,s\right)dsd\tau \\ \le 2\sum _{i=0}^{m}{I}_{\theta }^{\tau }{p}_{i}\left(t,x\right).\end{array}$
(10)

By using (10) and the fact that ${I}_{\theta }^{r}{p}_{i}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\to 0$as t → ∞; i = 0,..., m we deduce that

$\underset{t\to \infty }{\text{lim}}\left|u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)-{u}_{0}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\right|=0.$

Consequently, all solutions of problem (1)-(3) are uniformly locally attractive.

4. An example

As an application and to illustrate our results, we consider the following system of delay integro-differential equations of fractional order

$c{D}_{\theta }^{r}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}{I}_{0,x}^{{r}_{2}}u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right),\phantom{\rule{2.77695pt}{0ex}}u\left(t-1,\phantom{\rule{2.77695pt}{0ex}}x-\frac{1}{4}\right),\phantom{\rule{2.77695pt}{0ex}}u\left(t-\frac{2}{3},\phantom{\rule{2.77695pt}{0ex}}x-\frac{1}{5}\right)\right)\phantom{\rule{2.77695pt}{0ex}};$
$\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in J:=\left[0,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[0,1\right],$
(11)
$u\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)={e}^{-t};\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \stackrel{̃}{J}:=\left[-1,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[-\frac{1}{4},1\right]\\left(0,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left(0,1\right],$
(12)
$\left\{\begin{array}{cc}\hfill u\left(t,\phantom{\rule{2.77695pt}{0ex}}0\right)={e}^{-t};\hfill & \hfill t\in \left[0,\phantom{\rule{2.77695pt}{0ex}}\infty \right),\hfill \\ \hfill u\left(0,\phantom{\rule{2.77695pt}{0ex}}x\right)=1;\hfill & \hfill x\in \left[0,1\right],\hfill \end{array}\right\$
(13)

where $r=\left({r}_{1},\phantom{\rule{2.77695pt}{0ex}}{r}_{2}\right)=\left(\frac{1}{4},\frac{1}{2}\right)$ and

$\left\{\begin{array}{cc}\hfill f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w\right)=\frac{x{t}^{\frac{-3}{4}}\left(|v|\text{sin}t+|w|{e}^{-\frac{1}{t}}\right)}{2+|u|+|v|+|w|};\hfill & \hfill \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left(0,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[0,1\right]\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w\in ℝ,\hfill \\ \hfill f\left(t,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w\right)=0;\hfill & \hfill \left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left\{0\right\}×\left[0,1\right]\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}u,\phantom{\rule{2.77695pt}{0ex}}v,\phantom{\rule{2.77695pt}{0ex}}w\in ℝ.\hfill \end{array}\right\$

We have for each x [0, ∞), μ(t) = e-t→ 0 as t → ∞.

Let us notice that the function f satisfies assumption (H), where

${p}_{0}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in {ℝ}_{+}×\left[0,1\right],$
$\left\{\begin{array}{c}\hfill {p}_{1}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=x{t}^{-\frac{3}{4}}|\text{sin}t|;\phantom{\rule{1em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left(0,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[0,1\right],\hfill \\ \hfill {p}_{1}\left(0,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x\in \left[0,1\right],\hfill \end{array}\right\$

and

$\left\{\begin{array}{c}\hfill {p}_{2}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=x{t}^{-\frac{3}{4}}{e}^{-\frac{1}{t}};\phantom{\rule{1em}{0ex}}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)\in \left(0,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[0,1\right],\hfill \\ \hfill {p}_{2}\left(0,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x\in \left[0,1\right].\hfill \end{array}\right\$

Also, for each x 0[1], we get

$\begin{array}{l}\hfill {I}_{\theta }^{r}{p}_{0}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=0,\\ \hfill {I}_{\theta }^{r}{p}_{1}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)& =\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}{p}_{1}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau \phantom{\rule{2em}{0ex}}\\ =\frac{1}{\Gamma \left(\frac{1}{4}\right)\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{-\frac{3}{4}}{\left(x-s\right)}^{-\frac{1}{2}}s{\tau }^{-\frac{3}{4}}|\text{sin}\tau |dsd\tau \phantom{\rule{2em}{0ex}}\\ \le \frac{1}{\Gamma \left(\frac{1}{4}\right)\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{-\frac{3}{4}}{\left(x-s\right)}^{-\frac{1}{2}}s{\tau }^{-\frac{3}{4}}dsd\tau \phantom{\rule{2em}{0ex}}\\ =\frac{\Gamma \left(\frac{1}{4}\right)\Gamma \left(2\right)}{\Gamma \left(\frac{5}{4}\right)\Gamma \left(\frac{5}{2}\right)}{t}^{-\frac{1}{2}}{x}^{\frac{3}{2}}\to 0\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}t\to \infty ,\phantom{\rule{2em}{0ex}}\end{array}$

and

$\begin{array}{ll}\hfill {I}_{\theta }^{r}{p}_{2}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)& =\frac{1}{\Gamma \left({r}_{1}\right)\Gamma \left({r}_{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{{r}_{1}-1}{\left(x-s\right)}^{{r}_{2}-1}{p}_{2}\left(\tau ,\phantom{\rule{2.77695pt}{0ex}}s\right)dsd\tau \phantom{\rule{2em}{0ex}}\\ =\frac{1}{\Gamma \left(\frac{1}{4}\right)\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{\frac{-3}{4}}{\left(x-s\right)}^{\frac{-1}{2}}s{\tau }^{\frac{-3}{4}}{e}^{-\frac{1}{\tau }}dsd\tau \phantom{\rule{2em}{0ex}}\\ \le \frac{1}{\Gamma \left(\frac{1}{4}\right)\Gamma \left(\frac{1}{2}\right)}\underset{0}{\overset{t}{\int }}\underset{0}{\overset{x}{\int }}{\left(t-\tau \right)}^{\frac{-3}{4}}{\left(x-s\right)}^{\frac{-1}{2}}s{\tau }^{\frac{-3}{4}}dsd\tau \phantom{\rule{2em}{0ex}}\\ =\frac{\Gamma \left(\frac{1}{4}\right)\Gamma \left(2\right)}{\Gamma \left(\frac{5}{4}\right)\Gamma \left(\frac{5}{2}\right)}{t}^{\frac{-1}{2}}{x}^{\frac{3}{2}}\to 0\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}t\to \infty .\phantom{\rule{2em}{0ex}}\end{array}$

Thus

$\underset{t\to \infty }{\text{lim}}{I}_{\theta }^{r}{p}_{i}\left(t,\phantom{\rule{2.77695pt}{0ex}}x\right)=0;x\in \left[0,1\right];i=0,1,2.$

Hence by Theorem 3.3, the problem (11)-(13) has a solution defined on $\left[-1,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[-\frac{1}{4},1\right]$ and all solutions are uniformly locally attractive on $\left[-1,\phantom{\rule{2.77695pt}{0ex}}\infty \right)×\left[-\frac{1}{4},1\right]$

References

1. 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.

2. Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, New Jersey; 2000.

3. Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific Publishing, New York; 2012.

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

5. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

6. 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.2009.05.001

7. Abbas S, Benchohra M: Darboux problem for implicit impulsive partial hyperbolic differential equations. Electron J Diff Equ 2011, 2011: 15.

8. 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.

9. Vityuk AN, Golushkov AV: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil 2004, 7: 318–325. 10.1007/s11072-005-0015-9

10. 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.2007.07.038

11. 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.2010.09.004

12. 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.2011.48.3.539

13. 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.2008.03.033

14. Dhage BC: Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness. Diff Equ Appl 2010, 2: 299–318.

15. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory Appl Gordon Breach Yverdon 1993.

16. Corduneanu C: Integral Equations and Stability of Feedback Systems. Acedemic Press, New York; 1973.

17. Granas A, Dugundji J: Fixed Point Theory. Springer-Verlag New York; 2003.

Acknowledgements

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.

Author information

Authors

Corresponding author

Correspondence to Dumitru Baleanu.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

SA wrote the first draft of the article, MB has corrected this draft, DB had prepared the final version of the study and enhanced the revised version of the manuscript. The authors read carefully and approved the final version of the manuscript.

Rights and permissions

Reprints and permissions

Abbas, S., Baleanu, D. & Benchohra, M. Global attractivity for fractional order delay partial integro-differential equations. Adv Differ Equ 2012, 62 (2012). https://doi.org/10.1186/1687-1847-2012-62