SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Existence of integro-differential solutions for a class of abstract partial impulsive differential equations

Marcos N Rabelo1*, Marcos Henrique2 and Giovana Siracusa1

Author Affiliations

1 Departamento De Matemática, Universidade Federal De Pernambuco, Recife-pe, Cep 50540-740, Brazil

2 Faculdade De Ciência E Tecnologia De Caruaru, Universidade De Pernambuco, Caruaru-pe, Cep 55002, Brazil

For all author emails, please log on.

Journal of Inequalities and Applications 2011, 2011:135  doi:10.1186/1029-242X-2011-135

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/135


Received:1 March 2011
Accepted:7 December 2011
Published:7 December 2011

© 2011 Rabelo et al; licensee Springer.

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.

Abstract

In this study, we investigate the existence of integro-differential solutions for a class of abstract partial impulsive differential equations.

Keywords:
integro-differential equations; neutral differential equations; analytic semigroup of compact operators; non-autonomous operators; family of evolution operators; mild solutions

1. Introduction

In this study, we established two existence results of solutions for a class of impulsive functional differential equations which can be described in the following form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M3">View MathML</a>

(1.3)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M4">View MathML</a> is a family of unbounded linear closed operators such that for each t ∈ [0, b], A(t) is the infinitesimal generator of analytic semigroup of linear bounded operators (St(s))s ≥ 0 on a Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>, endowed with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M6">View MathML</a>; the history <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M7">View MathML</a> is defined as ut(θ) = u(t + θ), θ ≤ 0; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a> is a set of measurable functions φ : (-∞, 0] → X endowed with appropriate seminorm; the operator D(t, ϕ) is defined as D(t, ϕ) = ϕ(0) + g(t, ϕ), where the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M9">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M10">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M11">View MathML</a>, i ∈ ℤ are appropriate functions for all i ∈ {1, ..., m}; 0 < t1 < < tm < b is a sequence of fixed real numbers and the symbol Δξ(t) represents the jump of the function ξ at the moment t, this means that Δξ (t) = ξ (t+) - ξ(t-), where the notation ξ(t+) and ξ(t-) represent, respectively, the right and the left-hand side limits of the function ξ at t.

There are many physical phenomena that are described by means of impulsive differential equations, for instance, biological systems, electrical engineering, chemical reactions, among others can be modeled by impulsive differential equations, a good survey on impulsive differential equations can be found in [1] see also [2,3]. However, impulsive actions can influence the behavior of solutions making the analysis more difficult. Motivated by this facts, the studies of such systems have drawn the attention of many researchers during last years.

Recently, Park et al. [4] have investigated the problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M12">View MathML</a>

(1.4)

In this model, the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M13">View MathML</a> is the infinitesimal generator of a compact analytic semigroup of bounded linear operators (T(t))t ≥ 0 on a Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a> such that 0 ∈ ρ(A), where ρ(A) is the resolvent set of the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M14">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M16">View MathML</a>, i = 1, 2,..., m are given functions that satisfy suitable conditions. Using the theory of fractional powers and priori estimates for compact operators, the authors established some existence result for the problem (1.4). Lately, Balachandran and Annapoorani [5] investigated the following class of abstract problem (1.5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M17">View MathML</a>

In the system (1.5), it was assumed that for each t ∈ [0, b] the operator A(t) is the infinitesimal generator of compact analytic semigroup of bounded linear operators on a Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. Moreover, the domain, D(A(t)), of the operators A(t) is assumed to be independent of t ∈ [0, b] and dense in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M18">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M19">View MathML</a>. To get their results, the authors used the conditions of Acquistapace and Terreni, see [6], to guarantee the existence of an evolution family of operators associated with the non-autonomous abstract Cauchy problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M20">View MathML</a>

Then, using fractional powers and operators theory the authors get some existence result based on a priori bounded estimates for compact operators.

Other authors have studied problems involving impulsive act, for retarded and neutral functional differential equations we cite [7-17], for applications of impulsive differential equations on biology and neural networks we cite [18-21]. On the other hand impulsive fractional differential equations is a topic treated in [22,23].

In this article, the study of a class of neutral impulsive integro-differential equations is proposed. To get our results, we used the technique involving the fixed point theory of compact and condensing operators. We pointed out that the problem studied in this article has not been considered in the literature, once that the approach used in this study is totally different from those studies mentioned above. Actually, the main difference is that in our study we need to use an assumptions of compactness on the nonlinear equation, and in applications, these assumptions make all differences, because, even in infinite dimensional Hilbert space it is not straightforward handedly with compact sets. However, in our applications, we overcome this difficult using a well-known criterion of compactness in Lp(Ω) space [[24], Kolmogorov-Riesz-Weil theorem]. This is the principal motivation of this study.

We now turn to a summary of this study. The second section provides tools which are necessary to establish the main results that are the Theorems 2.3 and 2.4. In third section, we apply our abstract results in concrete examples.

2. Preliminaries

In this study, the symbols <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M22">View MathML</a> stand for Banach spaces with their, respectively, norms and we denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M23">View MathML</a> the Banach space of bounded linear operators from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M24">View MathML</a> into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a> endowed with the uniform operator topology; particularly, we denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M25">View MathML</a> when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M26">View MathML</a>. we start defining the evolution operator associated with the family A(t), t ∈ [0. b].

Definition 2.1. A family of operators U(t, s), t s, t, s I is said to be an evolution family associated to the problem (2.1) if the following conditions hold:

(a) U(t, s)U(s, r) = U(t, r) for all r s t.

(b) For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M27">View MathML</a>, the function (t, s) → U(t, s)x is continuous from {(t, s), t s, t, s I} into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>.

(c) For each t > s, the function t U(t, s) is continuous differentiable with respect to t and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M28">View MathML</a>.

The family of evolution system U(t, s) is called exponential stable if there are positive constants <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M29">View MathML</a> and α such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M30">View MathML</a>, for every t, s ∈ [0, b].

Throughout this study, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M31">View MathML</a> denotes a family of unbounded closed linear operators defined in a common domain D, which is independent of t and dense in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. Moreover, we assume that the system

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M32">View MathML</a>

(2.1)

has an associated evolution family of operators U(t, s), t s, t, s I. For additional details and more properties about the family U(t, s), we refer the reader to [6,25,26].

To study the problem (2.1), we consider the space of normalized piecewise continuous functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33">View MathML</a>, this means that, a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M34">View MathML</a> belongs to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33">View MathML</a> if u is continuous at t ti, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M35">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M36">View MathML</a>, for all i = 1,..., m. It is well known that if it is equipped with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M37">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M33">View MathML</a> became a Banach space.

The technique used in this study is based on the compactness criterion. For this reason, we will make the following assumptions.

Put t0 = 0, tn+1 = τ and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M38">View MathML</a> we denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M39">View MathML</a>, the function given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M40">View MathML</a>

In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M41">View MathML</a> stands the set defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M42">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M43">View MathML</a>.

Lemma 2.1. ( [12]) A set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M44">View MathML</a> is relatively compact in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M45">View MathML</a> if and only if the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M46">View MathML</a> is relatively compact in the space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M47">View MathML</a>, for every i = 0, 1,..., n.

The next step is to define the phase space. This will be done in the following way. The space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a>, will be formed by all measurable functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M48">View MathML</a> with seminorm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M49">View MathML</a>. On the phase space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a> we assume the following condition. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M50">View MathML</a>, b > 0 be a function such that x0 = φ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M51">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M52">View MathML</a>. Then the following properties hold true.

(i) xt is in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M53">View MathML</a>;

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M54">View MathML</a>, where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K(·) is continuous, M(·) is locally bounded and H, K, M are independent of x(·).

Remark 2.1. To treat retarded impulsive differential equation we suitable modified the axioms of the abstract phase space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a>. Actually, we drop the condition of continuity of the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a>-valued function t xt, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M55">View MathML</a> in not a continuous function.

Following the ideas of [15], we used the notations <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M56">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M57">View MathML</a> which is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M58">View MathML</a>

In what following we give some examples of phase spaces whose the above axioms are satisfied.

Example 2.1. Consider the function g(θ) = eγθ, θ ≤ 0, γ ≥ 0, and let L2([0, π], ℝ) be the space of square integrable Lebesgue measure functions endowed with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M59">View MathML</a>. Then we define the phase space norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a> as being

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M60">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a> is endowed with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M61">View MathML</a>, for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M62">View MathML</a> then it is well known that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M63">View MathML</a> is a phase space and the conditions (i)-(iii) are fulfilled. In these particular example it is possible to show that H = 1, K(t) = 1 and M(t) = e-γt, for all t ≥ 0.

Motivated by Pazy [16,25] we adopt the following concept of mild solution to problem (1.1)-(1.3).

Definition 2.2. A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M64">View MathML</a>, is a local mild solution of problem (1.1)-(1.3) if the following conditions holds.

(i) u0 = ϕ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M65">View MathML</a>;

(ii) the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M66">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M67">View MathML</a>, for all i = 1,..., m;

(iii) the integral equation below is satisfied,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M68">View MathML</a>

is satisfied.

The tools used in this study are based on point fixed theory. For this reason, the next two theorems play important role in the development of our results.

Theorem 2.1. [[27], Leray-Schauder Alternative] Let C be a convex subset of a Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>, and assume 0 ∈ C. Let F : C C be a completely continuous operator, and let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M69">View MathML</a>

Then either <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M70">View MathML</a> is unbounded or F has a fixed point.

Theorem 2.2. [[28], Corollary 4.3.2] Suppose that D is a closed bounded convex subset of the Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a> and that B and C are continuous function from D to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a> with

(ma1) Bx + Cx D, for all x D.

(ma2) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M71">View MathML</a>is compact set; and

(ma3) there is a number 0 ≤ γ < 1 such that || Bx - By || ≤ γ || x - y ||, for all x, y D.

Then there is z D such that Bz + Cz = z.

Next, we stated some important conditions used in the proof of our results.

(H1) The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M72">View MathML</a> satisfy the following condition

(H1.1) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M73">View MathML</a> and consider the extension <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M74">View MathML</a> of ϕ which is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M75">View MathML</a>

(2.2)

Then, each bounded set B of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M76">View MathML</a> the family of functions {g(t, yt + ut), t ∈ [0, b], u B} is equi-continuous.

(H1.2) There are constants c1 and c2 such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M77">View MathML</a>, for all t ≥ 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M78">View MathML</a>

(H2) The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M79">View MathML</a> satisfies the following conditions.

(H2.1) The function (x, ϕ) → f(t, ϕ, x) is continuous for almost everywhere t ∈ [0, b].

(H2.2) The function t f(t, ϕ, x) is strong measurable for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M80">View MathML</a>.

(H2.3) There is a positive continuous function m : [0, b] → [0, ∞) and a nondecreasing positive continuous function ψ: ℝ → [0, ∞) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M81">View MathML</a>

for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M82">View MathML</a>

(H3) The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M83">View MathML</a> satisfy the following conditions.

(H3.1) The function ϕ e(t, s, ϕ) is continuous almost everywhere for all t, s ∈ [0, b].

(H3.2) The function (t, s) → e(t, s, ϕ) is strong measurable for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M84">View MathML</a>

(H3.3) There is a positive continuous function p : [0, b] → [0, ∞) and a nondecreasing integrable positive function Ω: ℝ → [0, ∞) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M85">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M86">View MathML</a>

(H4) For each function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M87">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M89">View MathML</a>, τ e(s, τ, uτ), τ ∈ [0,b] and t f(t, ut, x), t ∈ [0, b] are measurable functions for almost everywhere s ∈ [0, b] and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M90">View MathML</a>

Now we are already to state and prove the main result of this article.

Theorem 2.3. Assume that the conditions (H1) - (H4) are satisfied. In addition, suppose that the following assumptions hold.

(t1) The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M91">View MathML</a> is completely continuous.

(t2) The operators Ii are completely continuous and there are positive constants, Li such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M92">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M93">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M94">View MathML</a>, and i = 1,..., m.

(t3) For each bounded subsets <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M95">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M96">View MathML</a> the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M97">View MathML</a>

is relatively compact for each t s, t, s ∈ [0, b], where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M98">View MathML</a> is an extension of x in such manner that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M99">View MathML</a>, t ≤ 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M100">View MathML</a>, t ∈ [0, b].

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M101">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M102">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M103">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M104">View MathML</a>

then, the problems (1.1)-(1.3) have a mild solution.

Proof. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M105">View MathML</a> is a solution of (1.1)-(1.3) and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M106">View MathML</a> be a continuous extension of ϕ given in (H1). If we written the solution u(·) of the problem (1.1)-(1.3) as u(t) = x(t) + y(t), t ∈ (-∞, b], then we can see that x(t) = 0, t ≤ 0 and for t ∈ [0, b] the following integral equation hold true

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M107">View MathML</a>

Motivated by this remark we consider the space

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M108">View MathML</a>

endowed with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M109">View MathML</a>. Moreover, on Λ we define the operators Γi : Λ → Λ, i = 1, 2, 3 given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M110">View MathML</a>

for all t ∈ [0, b]. Using the fact that (U(t, s))t s is a evolution family of operators and assuming the conditions on f, g and the family of operator Ii, i = 1,..., m, it is not difficult see that t → Γi(t), t ∈ [0, b] is a normalized piecewise continuous function for all i = 1,..., m. This shows that Γ is well defined. In the next, we prove that the operator Γ = Γ1 + Γ2 + Γ3 satisfies all conditions of Theorem 2.1. As the proof is very long we split it into various steps.

Step 1. The operator 1 is completely continuous

Let xn ∈ Λ, n ∈ ℕ be a sequence of elements of Λ such that xn x as n → ∞ for some x ∈ Λ. From the boundedness of operators U(t, s) and the axioms of the phase space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a> it is easy to see that the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M111">View MathML</a> is bounded in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M8">View MathML</a>, which implies from condition (H1) the uniformity convergence of

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M112">View MathML</a>

on [0, b]. Thus, we have the continuity of Γ1. From condition (H1), and the axiom (iii) follows immediately the Γ1 applies bounded sets of Λ into equi-continuous sets of Λ. On the other hand, again by (iii), and using the fact that f is a completely continuous function, soon as infers that, for each t ∈ [0, b], the set {g(t, xt + yt), x B} is compact in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. The proof that Γ1 is a completely continuous operator is complete.

Step 2. The operator Γ2 is complete continuous

The condition (H3.1) permit us conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M113">View MathML</a> as n → ∞ almost everywhere for t, s ∈ [0, b]. By (H3.3), and the Lebesgue's dominated convergence theorem we conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M114">View MathML</a>

uniformly for t ∈ [0, b]. From the strong continuity of the operators (U(t, s))t s, we can conclude that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M115">View MathML</a>

as n → ∞, uniformly for t ∈ [0, b]. This fact and the properties of the evolution family U(t, s) lead us to the continuity of the operator Γ2. Next, we show that Γ2 takes bounded sets into equi-continuous sets. First, we observe from conditions (H2.3), (H3.3) and the axioms of phase space that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M116">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M117">View MathML</a>

(2.3)

are bounded sets in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>.

Let ε > 0 be the arbitrary positive real number and t1, t2 ∈ [0, b], t1 > t2. Thus, take into account the previous notes and using the assumption (iii) we see that the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M118">View MathML</a>

is relatively compact in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. Thus we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M119">View MathML</a>

thus, from the continuity of U(t, s) and the assumptions of compactness contained on the condition (t3) we can infer the existence of 0 < δ < ε such that if |t1 - t2| < δ then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M120">View MathML</a>

This shows the equi-continuity of Γ2. In what follows, we show that for each t ∈ [0, b] the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M121">View MathML</a>

(2.4)

where B ∈ Λ, is pre-compact in Λ. To do that, we observe from (2.3) that for each s ∈ [0, t] the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M122">View MathML</a>

is a bounded set. Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M123">View MathML</a>

which implies by [[28], Lemma 1.3]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M124">View MathML</a>

with diam(Cε) < ε, where diam(·) denotes the diameter of the set Cε and co {·} the convex hull. Taking all this into account we see that for each fixed t ∈ [0, b], the set Θ(t) in (2.4) is relatively compact set in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. This completes the proof that the operator Γ2 is completely continuous.

Step 3. The operator Γ3 is completely continuous

To show that is Γ3 is a completely continuous, consider a bounded subset B of Λ and for each i = 1,..., m, define the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M125">View MathML</a> as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M126">View MathML</a>

To prove that the sets <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M127">View MathML</a>, i = 1,..., m, are precompacts in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M128">View MathML</a>, consider t1, t2 ∈ (ti, ti+1], t1 > t2. Using the continuity of (t, s) → U (t, s)x, and the compactness of sets Ij(B), j = 1,..., m, given ε > 0 there is 0 < δ < ε such that if |t1 - t2| < δ we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M129">View MathML</a>

uniformly for x B. On the other hand for t ∈ (ti, ti+1) fixed, from our hypothesis it is not difficult see that the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M130">View MathML</a>

is relatively compact in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>

On the other hand, if t = ti, the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M131">View MathML</a> became

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M132">View MathML</a>

and proceeding as in the early case we infer that the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M133">View MathML</a> is relatively compact in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>, The prove that the set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M134">View MathML</a>

is an equi-continuous set of functions is done in the same manner as at the beginning

of the section. The proof that Γ3 is completely continuous is finished.

In the next, we obtain a priori estimative of the solutions for the equation λΓxλ = xλ, for λ ∈ (0, 1) and = Γ = Γ1 + Γ2 + Γ3. Let x be a solution of the equation λΓ(xλ) = xλ, in addition we use the notation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M135">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M136">View MathML</a>

this implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M137">View MathML</a>

If we take the right-hand side of the previous inequalities and call it of v(t) we have that mλ(t) ≤ v(t), for all t ∈ [0, b]. This leads us to the following inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M138">View MathML</a>

this yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M139">View MathML</a>

Next, we considered the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M140">View MathML</a>, thus we have that v(0) = ϖ(0) and v(t) ≤ ϖ(t), for all t ∈ [0, b], using this and the non-decreasingly properties of the function ψ(·), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M141">View MathML</a>

Observe that if we define the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M142">View MathML</a> then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M143">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M144">View MathML</a>

for all t ∈ [0, b]. Integrating the early inequality from 0 to t we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M145">View MathML</a>

The early inequalities enable us to conclude that the set {xλ, xλ = Γxλ, λ ∈ (0, 1)} is bounded. From Theorem 2.1 the problem (1.1)-(1.3) has a mild solution. The proof of theorem is completed.   □

In the next result, the following conditions are used.

(G1) There is a positive constant Lf such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M146">View MathML</a>

for every t ∈ [0, b] and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M147">View MathML</a>

(G2) There are positive constants di, i = 1,..., m, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M148">View MathML</a>

for every x,<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M149">View MathML</a>..

Theorem 2.4. Assume that the condition (H2)-(H3) and (G1)-(G2) are satisfied. In addition, suppose that the assumption (iii) of Theorem (2.3) is satisfied. Then if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M150">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M151">View MathML</a>

then the problem (1.1)-(1.3) has a mild solution.

Proof. Let us consider the operator Γ: Λ → Λ defined as in Theorem 2.3. We claim that there is r > 0 such that Γ(Br) ⊂ Br. Suppose by contradiction that this assumption is false. Then for each r > 0 there are tr ∈ [0, b] and ur(·) ∈ Br such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M152">View MathML</a> This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M153">View MathML</a>

take the lim inf in the previous inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M154">View MathML</a>

which is contradictory with our assumptions. So let r > 0 be such a number and consider the restriction <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M155">View MathML</a> of the operator Γ on Br, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M156">View MathML</a> Next, we split the operator Γ in the following way Γ = Γ1 + Γ2, where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M157">View MathML</a>

As shown in the proof of Theorem 2.3, it is not difficult to see that Γ2 is completely continuous and for u1, u2 ∈ Λ we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M158">View MathML</a>

The previous inequality shows that Γ1 is contractive. Now, by Theorem 2.2, we can conclude that the problem (1.1)-(1.3) has a mild solution.   □

3. Applications

The main aim of this section is to apply our abstract results in concrete examples. To this end, we handle with a very special kind of operators. To be more specific, on the Banach space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M159">View MathML</a> we define the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M160">View MathML</a> given by Ax(ξ) = x"(ξ), ξ ∈ [0, π] with domain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M161">View MathML</a>

It is well known that in this case A has a discrete spectrum which is given by -n2, n ∈ ℕ. Moreover, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a> has a completely orthonormal base formed by eigenfunctions of A associated with the eigenvalues -n2, which is given <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M162">View MathML</a>, n ∈ ℕ. This implies that the following conditions are satisfied.

(i) For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M163">View MathML</a>,<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M164">View MathML</a>,

(ii) For each f D(A), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M165">View MathML</a>,

where 〈·,·〉 represents the inner product in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M5">View MathML</a>. Taking into account all these information, it is possible to prove that the operator A is the infinitesimal generator of a compact semigroup of bounded linear operators (T(t))t ≥ 0, which is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M166">View MathML</a>

To guarantee the existence of an evolution family associated with the problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M167">View MathML</a>

(3.1)

the following assumptions on the function a0 : [0, b] × [0, π] → ℝ are made

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M212">View MathML</a> There are constants c > 0 and α ∈ (0, 1) such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M168">View MathML</a>

for all t, s ∈ [0, b] and almost everywhere ξ ∈ [0, π].

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M213">View MathML</a> there is a real number c0 such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M169">View MathML</a>

for all τ ∈ [0, ∞) and ξ ∈ [0, π].

Letting D(A(t)) = D(A) for all t ≥ 0 and A(t)x(ξ) = a0(t, ξ)x"(ξ), ξ ∈ [0, π], we have that the system (3.1) has an associated evolution family of operators (U(t, s))t s which is given explicitly by the following formula:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M170">View MathML</a>

Using the properties of semigroup (T(t))t ≥ 0 it is straightforward to show that U(t, s) satisfies the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M171">View MathML</a>

Next, we consider the following partial differential equations

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M172">View MathML</a>

(3.2)

To model the problem (3.2) we choose as the phase space the set formed by all piecewise continuous functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M173">View MathML</a> which sups ≤ 0 h(θ) || φ(s) || < ∞, where h(θ) = eβθ, θ ≤ 0, and we denote this space by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M174">View MathML</a> equipping it with the norm <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M175">View MathML</a>. In order to show that the conditions (H1)-(H4) are satisfied we needed to consider the following assumptions.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M214">View MathML</a> The function P2 : (-∞, 0] × ℝ → [0, ∞) satisfies the following conditions:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M215">View MathML</a> for each η ∈ ℝ, s P2(s, η ) is a measurable and bounded function,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M216">View MathML</a> there is a positive constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M177">View MathML</a>

for all s ≤ 0, and η i ∈ ℝ, i = 1, 2.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217">View MathML</a> The functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M178">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M179">View MathML</a> are bounded almost everywhere on [-b, 0] × [0, π], <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M180">View MathML</a> is integrable on the interval [-b, 0].

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M218">View MathML</a>k2(·) ∈ L((-∞, π]) and s P2(s, η ) is measurable and bounded function for each η ∈ ℝ. In addition we assume the existence of positive constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M181">View MathML</a> such that the following inequality hold true

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M182">View MathML</a>

for almost everywhere s ∈ (-∞, 0] and η i ∈ [0, π], i = 1, 2.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M219">View MathML</a> The function P3 : [-π, ∞) × ℝ → ℝ satisfies the following conditions.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M220">View MathML</a> for each η ∈ ℝ, s P3(s, η ), s ∈ [-∞, b), is a measurable and bounded function,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M221">View MathML</a> there is a positive constant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M183">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M184">View MathML</a>

for all s ∈ [-π, ∞) and η i ∈ ℝ, i = 1, 2.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M222">View MathML</a> The function k3(·) ∈ L([-π, ∞)).

To transform the problem (3.2) into the abstract system (1.1), we define the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M185">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M186">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M187">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M188">View MathML</a>, i = 1, 2,..., n, respectively, given by,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M189">View MathML</a>

We shall show that the condition (H1) hold true. In fact, let x : (-∞, π] → L1(0, π) be a bounded function such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M190">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M191">View MathML</a>

The previous inequalities jointly with the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217">View MathML</a> show that the function t g(t, xt) is uniformly continuous on bounded subsets of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M192">View MathML</a>, L2(0, π)) which implies that the condition (H1.1) hold true. To prove that the condition (H1.2) is satisfied, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M193">View MathML</a>

which implies the condition (H1.2).

The next step is a proof that the function (x, ϕ) → f(t, ϕ, x) is continuous. However, with the help of condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M217">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M194">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M195">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M196">View MathML</a>, i = 1, 2. Thus we have shown that the condition (H2.1) is fulfilled. In particular, as P2 is continuous in the second variable we have that for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M197">View MathML</a> fixed the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M198">View MathML</a> is measurable. Thus from [[24], Theorem 1.2.1] we infer that t f(t, ϕ, x) is measurable for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M199">View MathML</a>. On the other hand, assuming that s P2(s, 0), s ∈ (-∞, 0] is bounded function, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M200">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M201">View MathML</a>

(3.3)

Thus the condition (H2.3) is fulfilled.

On the other hand, the same idea applied to prove that the previous functions is of Caratheádory type can be used to show that function e(·,·,·) satisfies the same property. Here it is mentioned that the functions that appear in the condition (H3.3) are given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M202">View MathML</a>

(3.4)

Finally, it remains that the condition (H4) is valid. However, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M203">View MathML</a>

where a function ϕ : (-∞, 0] → L2(0, π) is an element of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M204">View MathML</a>, L2(0, π)) if and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M205">View MathML</a>

with the norm defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M206">View MathML</a>, where μ = hdξ, and representing the Lebesgue measure on (-∞, 0]. Thus, following the ideas of [[29], Theorem 3.8] and using the fact that h(θ - t) ≤ G(-t)h(θ), θ ≤ 0, G(-t) = e-βt, t ≥ 0, we see that if u : (0, -∞] → L2(0, π) is admissible function in the sense of [29], then we derive the mensurability of t ut, t ∈ [0, b]. Thus, as e(·,·,·) and f(·,·,·) are measurable functions we infer that τ e(t, τ, uτ ) and τ f(t, uτ, x) for all t, τ ∈ [0, b], x L2(0, π). Now we will see that the conditions of the Theorem 2.3 hold. To see this, we observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M207">View MathML</a>

taking the advantage of the previous inequality we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M208">View MathML</a>

which implies from [[24], Theorem A.5.2] the assumption (t1).

The same idea which was used to prove the compactness of the function g can be used to prove that compactness of the operators Ii, i = 1,..., m. Regarding the inequality that appear in the condition (t2), we observe that to exhibit explicitly the ci constants, i = 1,..., m, the following account is necessary

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M209">View MathML</a>

for all i = 1,..., m.

As we choose the phase space as being <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M210">View MathML</a>, then it is possible to show that the constant H and the functions K(·) and M(·) that appear in the axioms (ii) and (iii) are given, respectively, by H = 1, K(t) = 1 and M(t) = e-βt, for all t ≥ 0.

Taking into account what was said before we derive the following result.

Theorem 3.1. Assume that all previous conditions are fulfilled. Assume in addition that the following inequalities hold,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/135/mathml/M211">View MathML</a>

where p(·) represents the right-hand side of the inequalities (3.4). Then the problem (3.2) has a mild solution.

4. Competing interests

The authors declare that they have no competing interests.

5. Authors' contributions

MNR conceived the study and participated in its design and coordination. MH participated in the design of the study and performed the typesetting of the text. GS participated in the design of the article. All authors read and approved the final manuscript.

References

  1. Bainov, DD, Pavel, S: Impulsive Differential Equations: Periodic Solutions and Applications. In: Pitman Monographs and Surveys in Pure and Applied Mathematics.p. x+228. Longman Scientific & Technical, Harlow, UK (1993)

  2. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)

  3. Lakshmikanthan, V, Bainov, D, Simeonov, P: Theory of Impulsive Differential Equations. In series in Modern Applied Mathematics. World Scientific, Teaneck, NJ. 6, (1989)

  4. Park, JY, Balachandran, K, Annapoorani, N: Existence results for impulsive neutral functional integrodifferential equations with infinite delay. Nonlinear Anal. 71, 3152–3162 (2009). Publisher Full Text OpenURL

  5. Balachandran, K, Annapoorani, N: Existence results for impulsive neutral evolution integrodifferential equations with infinite delay. Nonlinear Anal Hybrid Syst. 3, 674–684 (2009). Publisher Full Text OpenURL

  6. Schnaubelt, R: Asymptotic behavior of parabolic nonautonomous evolution equations. Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, pp. 401–472. Springer, Berlin (2004)

  7. Hernández, E, Prokopezyk, A, Ladeira, L: A note on partial functional differential equations with state-dependent delay. Nonlinear Anal Real World Appl. 7(4), 510–519 (2006). Publisher Full Text OpenURL

  8. Baghli, S, Benchohra, M: Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces. Elec J Differ Equ. 69, 1–19 (2008)

  9. Ahmad, S, Stamov, GT: Almost periodic solutions of N-dimensional impulsive competitive systems. Nonlinear Anal Real World Appl. 10(3), 1846–1853 (2009). Publisher Full Text OpenURL

  10. Ahmad, S, Stamov, GT: On almost periodic processes in impulsive competitive with delay and impulsive perturbations. Nonlinear Anal Real World Appl. 10(5), 2857–2863 (2009). Publisher Full Text OpenURL

  11. Hernádez, E, Henráquez, H: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. J Math Anal Appl. 221(1), 499–522 (1998)

  12. Hernïández, E, Pierri, M, Gonáalves, G: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Comput Math Appl. 52(3-4), 411–420 (2006). Publisher Full Text OpenURL

  13. Hernández, E, Rabelo, M, Henráquez, H: Existence of solutions for impulsive partial neutral functional differential equations. J Math Anal Appl. 331, 1135–1158 (2007). Publisher Full Text OpenURL

  14. Chang, YK, Anguraj, A, Arjunan, MM: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Anal Hybrid Syst. 2, 209–218 (2008). Publisher Full Text OpenURL

  15. Cuevas, C, Hernández, E, Rabelo, M: The existence of solutions for impulsive neutral functional differential equations. Comput Math Appl. 58(4), 744–757 (2009). Publisher Full Text OpenURL

  16. Hernández, E, Hernráquez, HR: Impulsive partial neutral differential equations. Appl Math Lett. 19, 215–222 (2006). Publisher Full Text OpenURL

  17. Hernádez, E, McKibben, M, Henráquez, HR: Existence results for abstract impulsive second order neutral functional differential equations. Nonlinear Anal Theory Methods Appl. 70(7), 2736–2751 (2009). Publisher Full Text OpenURL

  18. Stamov, GT: On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model. Appl Math Lett. 22(4), 516–520 (2009). Publisher Full Text OpenURL

  19. Stamov, GT: Almost periodic models of impulsive Hopfield neural networks. J Math Kyoto Univ. 49(1), 56–67 (2009)

  20. Stamov, GT: Stability processes of moving invariant manifolds in uncertain impulsive differential-difference equations. Math Bohem. 134(1), 67–76 (2009)

  21. Stamov, GT: Existence of almost periodic solutions for impulsive cellular neural networks. Rocky Mountain J Math. 38(4), 1271–1284 (2008). Publisher Full Text OpenURL

  22. Benchohra, M, Slimani, BA: Existence and uniqueness of solutions to impulsive fractional differential equations. Elec J Differ Equ. 2009(10), 11 (2009)

  23. Agarwal, RP, Benchohra, M, Slimani, BA: Existence results for differential equations with fractional order and impulses. Mem Diff Equ Math Phys. 44, 1–21 (2008)

  24. Vrabie, II: C0-semigroups and applications. North-Holland Mathematics Studies.p. xii+373. North-Holland Publishing Co., Amsterdam (2003)

  25. Pazy, A: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, Springer, New York (1983)

  26. Lunardi, A: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, Birkháauser Verlag, Basel (1995)

  27. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)

  28. Martin, RH Jr.: Nonlinear Operators and Differential Equations in Banach Spaces. Pure and Applied Mathematics,p. xi+440. Wiley, New York (1976)

  29. Hino, Y, Murakami, S, Naito, T: Functional-Differential Equations with Infinite Delay. Lectures Notes in Mathematics. Spring, New York (2003)