Open Access

Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems

Advances in Difference Equations20112011:5

DOI: 10.1186/1687-1847-2011-5

Received: 15 December 2010

Accepted: 24 May 2011

Published: 24 May 2011

Abstract

In this article, sufficient conditions for the existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions in Banach spaces have been presented using fractional calculus, resolvent operators, and Banach fixed point theorem. As an application that illustrates the abstract results, a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given.

AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05.

Keywords

Fractional integrodifferential systems resolvent operators nonlocal and impulsive conditions fixed point theorem

Introduction

Many fractional models can be represented by the following system
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ3_HTML.gif
(1.3)

in a Banach space X, where 0 < α ≤ 1, t [0, a], u 0 X, i = 1, 2,..., m and 0 < t 1 < t 2 < ··· < t m < a. We assume that -A(t,.) is a closed linear operator defined on a dense domain D(A) in X into X such that D(A) is independent of t. It is assumed also that -A(t,.) generates an evolution operator in the Banach space X. The functions f : J X r+1X, g : Λ × X k+1X, h : PC(J, X) → X, u(β) = (u(β 1),..., u(β r )), u(γ) = (u(γ 1),..., u(γ k )), and β p , γ q : JJ are given, where p = 1, 2,..., r and q = 1, 2,..., k. Here J = [0, a] and Λ = {(t, s). 0 ≤ sta}. Let PC (J, X) consist of functions u from J into X, such that u(t) is continuous at tt i and left continuous at t = t i and the right limit https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq1_HTML.gif exists for i = 1, 2,..., m. Clearly PC(J, X) is a Banach space with the norm ||u|| PC = sup tJ ||u(t)||, and let https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq2_HTML.gif constitutes an impulsive condition. Fractional differential equations have proved to be valuable tools in the modelling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [15]). They involve a wide area of applications by bringing into a broader paradigm concepts of physics and mathematics [68]. There has been a significant development in fractional differential and partial differential equations in recent years, see Kilbas et al. [9, 10], also in fractional nonlinear systems with delay and fractional variational principles with delay, see Baleanu et al. [11, 12].

The existence results to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [13, 14], subsequently, many authors were pointed in the same field, see reference therein. Deng [15] indicated that, using the nonlocal condition u(0) + h(u) = u 0 to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem u(0) = u 0. Let us observe also that since Deng's papers, the function h is considered
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equa_HTML.gif

where c k , k = 1, 2,..., p are given constants and 0 ≤ t 1 < ··· < t p a. However, among the previous research on nonlocal cauchy problems, few are concerned with mild solutions of fractional semilinear differential equations, see Mophou and N'Guérékata [16], and others with fractional nonlocal boundary value problems, for instance, Ahmad et al. [17, 18].

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years, because all the structures of its emergence have deep physical background and realistic mathematical model. The theory of impulsive differential equations appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. It has seen considerable development in the last decade, see the monographs of Bainov and Simeonov [19], Lakshmikantham et al. [20], and Samoilenko and Perestyuk [21] where numerous properties of their solutions are studied, and detailed bibliographies are given.

Recently, the existence of solutions of fractional abstract differential equations with nonlocal initial condition was investigated by N'Guérékata [22] and Li [23]. Much attention has been paid to existence results for the impulsive differential and integrodifferential equations of fractional order in abstract spaces, see Benchohra et al. [2, 24]. Several authors have studied the existence of solutions of abstract quasilinear evolution equations in Banach space [2527].

Regarding this article, it generalizes previous results concerned the existence of solutions to nonlocal and impulsive integrodifferential equations of quasilinear type with delays of arbitrary orders. Section "Preliminaries" is devoted to a review of some essential results. In next section, we state and prove our main results, the last section deals to giving an example to illustrate the abstract results.

1 Preliminaries

Let X and Y be two Banach spaces such that Y is densely and continuously embedded in X. For any Banach space Z, the norm of Z is denoted by ||·|| Z . The space of all bounded linear operators from X to Y is denoted by B(X, Y) and B(X, X) is written as B(X). We recall some definitions in fractional calculus from Gelfand-Shilov [28] and Podlubny [29], then some known facts of the theory of semigroups from Pazy [30].

Definition 2.1 The fractional integral of order with the lower limit zero for a function f C([0, ∞)) is defined as
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equb_HTML.gif
provided the right side is pointwise defined on [0, ∞), where Γ is the gamma function. Riemann-Liouville derivative of order α with the lower limit zero for a function f C([0, ∞)) can be written as
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equc_HTML.gif
The Caputo derivative of order for a function f C([0, ∞)) can be written as
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equd_HTML.gif
Remark 2.1
  1. (1)
    If f C 1([0, ∞)), then
    https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Eque_HTML.gif
     
  2. (2)

    The Caputo derivative of a constant is equal to zero.

     
  3. (3)

    If f is an abstract function with values in X, then integrals which appear in Definition 2.1 are taken in Bochner's sense.

     
Definition 2.2 A two parameter family of bounded linear operators U(t, s), 0 ≤ sta, on X is called an evolution system if the following two conditions are satisfied
  1. (i)

    U(t, t) = I, U(t, r)U(r, s) = U(t, s) for 0 ≤ srta,

     
  2. (ii)

    (t, s) → U(t, s) is strongly continuous for 0 ≤ sta.

     

More detail about evolution system and quasilinear equation of evolution can be found in [30, Chap. 5 and Sect. 6.4, respectively].

Let E be the Banach space formed from D(A) with the graph norm. Since - A(t) is a closed operator, it follows that - A(t) is in the set of bounded operators from E to X.

Definition 2.3[3133] A resolvent operators for problem (1.1)-(1.3) is a bounded operators valued function R u (t, s) B(X), 0 ≤ sta, the space of bounded linear operators on X, having the following properties:
  1. (i)

    R u (t, s) is strongly continuous in s and t, R u (s, s) = I, 0 ≤ sa, ||R u (t, s)|| ≤ Me N(t, s)for some constants M and N.

     
  2. (ii)

    R u (t, s)E E, R u (t, s) is strongly continuous in s and t on E.

     
  3. (iii)
    For x X, R u (t, s)x is continuously differentiable in s [0, a] and
    https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equf_HTML.gif
     
  4. (iv)
    For x X and s [0, a], R u (t, s)x is continuously differentiable in t [s, a] and
    https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equg_HTML.gif
     

with https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq4_HTML.gif are strongly continuous on 0 ≤ sta. Here R u (t, s) can be extracted from the evolution operator of the generator - A(t, u). The resolvent operator is similar to the evolution operator for nonautonomous differential equations in a Banach space. Let Ω be a subset of X.

Definition 2.4 (Compare [31] with [7, 22, 34]) By a mild solution of (1.1)-(1.3) we mean a function u PC(J : X) with values in Ω satisfying the integral equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ4_HTML.gif
(2.1)

for all u0 X.

Definition 2.5 (Compare [35, 36] with [2]) By a classical solution of (1.1)-(1.3) on J, we mean a function u with values in X such that:
  1. (1)

    u is continuous function on J \{t 1, t 2,..., t m } and u(t) D(A),

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq5_HTML.gif exists and continuous on J 0, 0 < α < 1,

     
  3. (3)

    u satisfies (1.1) on J 0, the nonlocal condition (1.2) and the impulsive condition (1.3), where J 0 = (0, a]\{t 1, t 2,..., t m }. We assume the following conditions

     
(H1) h : PC(J : Ω) → Y is Lipschitz continuous in X and bounded in Y , i.e., there exist constants k 1 > 0 and k 2 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equh_HTML.gif

For the conditions (H 2 ) and (H 3 ) let Z be taken as both × and Y.

(H2) g : Λ × Z k+1Z is continuous and there exist constants k 3 > 0 and k 4 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equi_HTML.gif
(H3) f : J × Z r+1Z is continuous and there exist constants k 5 > 0 and k 6 > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equj_HTML.gif

(H4) β p , γ q : JJ are bijective absolutely continuous and there exist constants c p > 0 and b q > 0 such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq7_HTML.gif , respectively, for t J, p = 1,..., r and q = 1,..., k.

(H5) I i : XX are continuous and there exist constants l i > 0, i = 1, 2,..., m such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equk_HTML.gif

Let us take M 0 = max ||R u (t, s)|| B(Z), 0 ≤ sta, u Ω.

(H6) There exist positive constants δ 1, δ 2, δ 3 (0, δ /3] and λ1, λ2, λ3 [0, https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq8_HTML.gif ) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equl_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equm_HTML.gif

where ρ = σ [k 5(1/c 1 + ··· +1/c r )+ k 3(1/b 1 + ··· +1/b k )], θ = σδ (k 3 + k 5)+ ρδ + σ (k 4 + k 6), https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq10_HTML.gif .

Main results

Lemma 3.1 Let R u (t, s) the resolvent operators for the fractional problem (1.1)-(1.3). There exists a constant K > 0 such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equn_HTML.gif

for every u, v PC(J : X) with values in Ω and every ω Y , see [30, lemma 4.4, p. 202].

Let S δ = {u : u PC(J : X), u(0) + h(u) = u 0, Δu(t i) = I i (u(t i )), ||u|| ≤ δ}, for t J, δ > 0, u 0 X and i = 1,..., m.

Lemma 3.2
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equo_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equp_HTML.gif
Proof We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equq_HTML.gif
Using H2, H3, and H4, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equr_HTML.gif

Hence the required result.

Theorem 3.3 Suppose that the operator -A(t, u) generates the resolvent operator R u (t, s) with ||R u (t, s)||≤ Me N(t-s). If the hypotheses (H1)-(H6) are satisfied, then the fractional integro-differential equation (1.1) with nonlocal condition (1.2) and impulsive condition (1.3) has a unique mild solution on J for all u 0 X.

Proof Consider a mapping P on S δ defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equs_HTML.gif
We shall show that P : S δ S δ . For u S δ , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equt_HTML.gif
Using H1, Lemma 3.2 and H5, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equu_HTML.gif
From assumption H6, one gets ||(Pu μ )(t)|| Y δ. Thus, P maps S δ into itself. Now for u, v S δ , we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equv_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equw_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equx_HTML.gif
Applying Lemma 3.1 and H1, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equy_HTML.gif
Also, we apply Lemmas 3.1,3.2, H2, H3, H4, and H6, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equz_HTML.gif
Again, Lemma 3.1, H5 and H6, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equaa_HTML.gif
It follows from these estimations that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equab_HTML.gif

where 0 ≤ λ < 1. Thus P is a contraction on S δ . From the contraction mapping theorem, P has a unique fixed point u S δ which is the mild solution of (1.1)-(1.3) on J.

Theorem 3.4 Assume that
  1. (i)

    Conditions (H1)-(H6) hold,

     
  2. (ii)

    Y is a reflexive Banach space with norm ||·||,

     
  3. (iii)

    The functions f and g are uniformly Hölder continuous in t J.

     

Then the problem (1.1)-(1.3) has a unique classical solution on J.

Proof From (i), applying Theorem 3.3, the problem (1.1)-(1.3) has a unique mild solution u S δ .. Set
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equac_HTML.gif
In order to prove the regularity of the mild solution, we use the further assumptions, it is easy to conclude that the function ω(t) is also uniformly Hölder continuous in t J. Consider the following fractional differential equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ5_HTML.gif
(3.1)

with the nonlocal condition (1.2) and impulsive condition (1.3).

According to Pazy [30], the late problem has a unique solution v on J intoX given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equad_HTML.gif

Noting that, each term on the right-hand side belongs to D(A), using the uniqueness of v(t), we have that u(t) D(A). It follows that u is a unique classical solution of (1.1)-(1.3) on J.

Application

Consider the nonlinear integro-partial differential equation of fractional order
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ6_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ7_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equ8_HTML.gif
(4.3)
where 0 < α ≤ 1, 0 ≤ t 1 < ··· < t p a, x R n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq12_HTML.gif , q= (q 1,...,q n ) is an n-dimensional multi-index, |q| = q 1 + ··· + q n , and w i , i = 1, 2, is given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equae_HTML.gif
Let L2(R n ) be the set of all square integrable functions on R n . We denote by C m (R n ) the set of all continuous real-valued functions defined on R n which have continuous partial derivatives of order less than or equal to m. By https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq13_HTML.gif we denote the set of all functions f C m (R n ) with compact supports. Let H m (R n ) be the completion of https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq13_HTML.gif with respect to the norm
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equaf_HTML.gif
It is supposed that
  1. (i)
    The operator https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq14_HTML.gif is uniformly elliptic on R n . In other words, all the coefficients a q , |q| = 2m, are continuous and bounded on R n and there is a positive number c such that
    https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equag_HTML.gif

    for all x R n and all ξ ≠ 0, ξ R n , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_IEq16_HTML.gif .

     
  2. (ii)

    All the coefficients a q , |q| = 2m, satisfy a uniform Hölder condition on R n . Under these conditions the operator A with domain of definition D(A) = H 2m (R n ) generates an evolution operator defined on L 2(R n ), and it is well known that H 2m (R n ) is dense in X = L 2(R n ) and the initial function g(x) is an element in Hilbert space H 2m (R n ), see [14, 15, 35]. Applying Theorem 3.3, this achieves the proof of the existence of mild solutions of the system (4.1)-(4.3). In addition,

     
  3. (iii)

    If the coefficients b q , c q , |q| ≤ 2m - 1 satisfy a uniform Hölder condition on R n and the operators F and G satisfy

     
There are numbers L 1, L 2 ≥ 0 and λ 1, λ 2 (0, 1) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equah_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2011-5/MediaObjects/13662_2010_Article_5_Equai_HTML.gif

for all t, s I, (t, η), (s, η) Δ, and all x R n . Applying Theorem 3.4, we deduce that (4.1)-(4.3) has a unique strong solution.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Guelma University

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