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Partial neutral functional integro-differential equations of fractional order with delay

Saïd Abbas1, Mouffak Benchohra2 and Alberto Cabada3*

Author affiliations

1 Laboratoire de Mathématiques, Université de Saïda, B.P. 138, Saïda, 20000, Algeria

2 Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, Sidi Bel-Abbès, 22000, Algeria

3 Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

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Citation and License

Boundary Value Problems 2012, 2012:128  doi:10.1186/1687-2770-2012-128

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/128


Received:9 July 2012
Accepted:24 October 2012
Published:6 November 2012

© 2012 Abbas 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 paper we obtain sufficient conditions for the existence of solutions of some classes of partial neutral integro-differential equations of fractional order by using suitable fixed point theorems.

MSC: 26A33.

Keywords:
integro-differential equation; left-sided mixed Riemann-Liouville integral of fractional order; Caputo fractional-order derivative; finite delay; infinite delay; solution; fixed point

1 Introduction

Fractional differential and integral equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Abbas et al.[1], Baleanu et al.[2], Kilbas et al.[3], Lakshmikantham et al.[4], Podlubny [5], and the references therein.

In [6], Czlapinski proved some results for the following system of the Darboux problem for the second-order partial functional differential equations of the form

(1)

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M8">View MathML</a>, and ℬ is a vector space of real-valued functions defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M9">View MathML</a>, equipped with a semi-norm and satisfying some suitable axioms, which was introduced by Hale and Kato [7]; see also [8-10] with rich bibliography concerning functional differential equations with infinite delay. Recently, Abbas et al. studied some existence results for the Darboux problem for several classes of fractional-order partial differential equations with finite delay [11,12] and others with infinite delay [13,14].

Motivated by the above papers, in this article we deal with the existence of solutions for two systems of neutral integro-differential equations of fractional order with delay. First, we consider the system of fractional-order neutral integro-differential equations with finite delay of the form

(3)

(4)

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M13">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M18">View MathML</a> is the left-sided mixed Riemann-Liouville integral of order r (see Section 2 for definition), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M19">View MathML</a> is the fractional Caputo derivative of order r, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M21">View MathML</a> are given continuous functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M24">View MathML</a> are given absolutely continuous functions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M26">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M28">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M29">View MathML</a> is the Banach space of continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30">View MathML</a> coupled with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M31">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M32">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M33">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M35">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M36">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M37">View MathML</a> represents the history of the state from time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M38">View MathML</a> up to the present time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M39">View MathML</a>.

Next, we consider the system of fractional-order neutral integro-differential equations with infinite delay of the form

(6)

(7)

(8)

where J, φ, ψ are as in the problem (3)-(5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M45">View MathML</a> are given continuous functions, and ℬ is called a phase space that will be specified in Section 4.

During the last two decades, many authors have considered the questions of existence, uniqueness, estimates of solutions, and dependence with respect to initial conditions of the solutions of differential and integral equations of two and three variables (see [15-19] and the references therein).

It is clear that more complicated partial differential systems with deviated variables and partial differential integral systems can be obtained from (3) and (6) by a suitable definition of f and g. Barbashin [20] considered a class of partial integro-differential equations which appear in mathematical modeling of many applied problems (see [21], Section 19). Recently Pachpatte [22,23] considered some classes of partial functional differential equations which occur in a natural way in the description of many physical phenomena.

We present the existence results for our problems based on the nonlinear alternative of the Leray-Schauder theorem. The present results extend those considered with integer order derivative [6,9,16,24,25] and those with fractional derivative [11,12,26].

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M46">View MathML</a> we denote the Banach space of all continuous functions from J into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M48">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M49">View MathML</a> denotes the usual supremum norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a>.

Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M51">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M52">View MathML</a>

As usual, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M53">View MathML</a> we denote the space of absolutely continuous functions from J into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M55">View MathML</a> is the space of Lebesgue-integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M56">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M57">View MathML</a>

Definition 2.1 ([27])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60">View MathML</a>. The left-sided mixed Riemann-Liouville integral of order r of u is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M61">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M62">View MathML</a> is the (Euler’s) gamma function defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M63">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M64">View MathML</a>.

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M65">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M66">View MathML</a>.

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M68">View MathML</a> exists for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M69">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M70">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M71">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M72">View MathML</a>

Example 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M74">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M75">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M76">View MathML</a> we mean <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M77">View MathML</a>.

Definition 2.3 ([27])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M60">View MathML</a>. The Caputo fractional-order derivative of order r of u is defined by the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M80">View MathML</a>

The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M66">View MathML</a> is included, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M82">View MathML</a>

Example 2.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M84">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M85">View MathML</a>

In the sequel, we need the following lemma.

Lemma 2.5 ([26])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M87">View MathML</a>. Then the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M88">View MathML</a>of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M89">View MathML</a>

is given by the following expression:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M90">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M91">View MathML</a>

As a consequence of Lemma 2.5, it is not difficult to verify the following result.

Corollary 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M93">View MathML</a>. A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M94">View MathML</a>is a solution of the problem (3)-(5) if and only ifusatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M95">View MathML</a>

Also, we need the following theorem.

Theorem 2.7 (Nonlinear alternative of Leray-Schauder type [28])

By<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M96">View MathML</a>and∂Uwe denote the closure ofUand the boundary ofUrespectively. LetXbe a Banach space andCa nonempty convex subset ofX. LetUbe a nonempty open subset ofCwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M98">View MathML</a>be a completely continuous operator.

Then either

(a) Thas fixed points or

(b) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M101">View MathML</a>.

3 Existence results with finite delay

Let us start by defining what we mean by a solution of the problem (3)-(5).

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102">View MathML</a> is said to be a solution of the problem (3)-(5) if u satisfies equations (3), (5) on J and the condition (4) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M103">View MathML</a>.

Further, we present conditions for the existence of a solution of the problem (3)-(5).

(H1) There exist nonnegative functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M104">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M105">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M108">View MathML</a>.

(H2) For any bounded set B in E, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M109">View MathML</a> is equicontinuous in E, and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M110">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M111">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M112">View MathML</a>

Theorem 3.2Assume that the hypotheses (H1) and (H2) hold. Then if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M113">View MathML</a>

(9)

the problem (3)-(5) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M114">View MathML</a>.

Proof Transform the problem (3)-(5) into a fixed point problem. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M115">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M116">View MathML</a>

(10)

It is clear that N maps E into itself. By Corollary 2.6, the problem of finding the solutions of the problem (3)-(5) is reduced to finding the solutions of the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M117">View MathML</a>. We shall show that the operator N satisfies all the conditions of Theorem 2.7. The proof will be given in two steps.

Step 1: Nis continuous and completely continuous.

Using (H2) we deduce that g is a complete continuous operator from E to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a>, so it suffices to show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M119">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M120">View MathML</a>

(11)

is continuous and completely continuous. The proof will be given in several claims.

Claim 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121">View MathML</a>is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M122">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M123">View MathML</a> in E. Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124">View MathML</a>, we have

Hence, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, we get

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M123">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M129">View MathML</a> and f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M130">View MathML</a> are continuous, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M131">View MathML</a>

Claim 2: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121">View MathML</a>maps bounded sets into bounded sets inE.

Indeed, it is enough to show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M133">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M134">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M135">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M136">View MathML</a>.

By (H2) and (H3), we have that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M138">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M139">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M140">View MathML</a>

Claim 3: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121">View MathML</a>maps bounded sets inEinto equicontinuous sets inE.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M133">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M147">View MathML</a>. Then

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M150">View MathML</a>, the right-hand side of the above inequality tends to zero with the same rate of convergence for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M151">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M147">View MathML</a>.

The equicontinuity for the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M156">View MathML</a> is obvious. As a consequence of Claims 1 to 3 together with the Arzelá-Ascoli theorem, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M121">View MathML</a> is continuous and completely continuous.

Step 2: A priori bounds.

We shall show that there exists an open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M158">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M159">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M102">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M163">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M164">View MathML</a>. Thus, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M166">View MathML</a>

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M167">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M168">View MathML</a>

It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M169">View MathML</a>

As consequence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M170">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M171">View MathML</a>.

On the contrary, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M172">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M173">View MathML</a>. So, from the previous inequalities and the condition (9), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M174">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M175">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M176">View MathML</a>

By our choice of U, there is no <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M99">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M163">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100">View MathML</a>.

As a consequence of Steps 1 and 2 together with Theorem 2.7, we deduce that N has a fixed point u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M96">View MathML</a> which is a solution to the problem (3)-(5). □

4 The phase space ℬ

The notation of the phase space ℬ plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato (see [7]). For further applications, see, for instance, the books [10,29,30] and their references. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M182">View MathML</a>. Furthermore, in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M184">View MathML</a>, we write simply ℰ. Consider the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M185">View MathML</a> a semi-normed linear space of functions mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M9">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a> and satisfying the following fundamental axioms which were adapted from those introduced by Hale and Kato for ordinary differential functional equations:

(A1) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M188">View MathML</a> is a continuous function on J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M189">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M190">View MathML</a>, then there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M191">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, the following conditions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M193">View MathML</a> is in ℬ;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M194">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M195">View MathML</a>.

(A2) For the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M196">View MathML</a> in (A1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M193">View MathML</a> is a ℬ-valued continuous function on J.

(A3) The space ℬ is complete.

Now, we present some examples of phase spaces [6,9].

Example 4.1 Let ℬ be the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198">View MathML</a> which are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M200">View MathML</a>, with the semi-norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M201">View MathML</a>

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M202">View MathML</a>. The quotient space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M203">View MathML</a> is isometric to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M204">View MathML</a> of all continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M47">View MathML</a> with the supremum norm. This means that partial differential functional equations with finite delay are included in our axiomatic model.

Example 4.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M207">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M208">View MathML</a> be the set of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198">View MathML</a>, for which a limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M210">View MathML</a> exists, with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M211">View MathML</a>

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M213">View MathML</a>.

Example 4.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M214">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M215">View MathML</a>

be the semi-norm for the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M216">View MathML</a> of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M198">View MathML</a> which are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M30">View MathML</a> measurable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M219">View MathML</a>, and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M220">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M221">View MathML</a>

5 Existence results with infinite delay

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M222">View MathML</a>

Let us start by defining what we mean by a solution of the problem (6)-(8).

Definition 5.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M223">View MathML</a> is said to be a solution of (6)-(8) if u satisfies equations (6) and (8) on J and the condition (7) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M224">View MathML</a>.

Now, we present conditions for the existence of a solution of the problem (6)-(8).

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M225">View MathML</a>) There exist nonnegative functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M226">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M227">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M230">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231">View MathML</a>) For any bounded set B in Ω, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M232">View MathML</a> is equicontinuous in Ω, and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M233">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M234">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M235">View MathML</a>

Theorem 5.2Assume that the hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M225">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231">View MathML</a>) hold. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M238">View MathML</a>

(12)

then the problem (6)-(8) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M239">View MathML</a>.

Proof Transform the problem (6)-(8) into a fixed point problem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M223">View MathML</a> and define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M241">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M242">View MathML</a>

(13)

As in Theorem 3.2, we can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243">View MathML</a> maps Ω into itself.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M244">View MathML</a> be a function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M245">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M246">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M247">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M248">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M249">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M247">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M251">View MathML</a> the function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M252">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M253">View MathML</a> satisfies the integral equation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M254">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124">View MathML</a>, we can decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M253">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M257">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M124">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M259">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M34">View MathML</a>, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M261">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M262">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M263">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M264">View MathML</a> be the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M266">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265">View MathML</a> is a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M264">View MathML</a>.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M269">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M270">View MathML</a>.

Let the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M271">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M272">View MathML</a>

(14)

Then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243">View MathML</a> has a fixed point in Ω if and only if P has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M265">View MathML</a>. As in the proof of Theorem 3.2, we can show that the operator P satisfies all the conditions of Theorem 2.7. Indeed, to prove that P is continuous and completely continuous and by using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M231">View MathML</a>), it suffices to show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M276">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M277">View MathML</a>

(15)

is continuous and completely continuous. Also, we can show that there exists an open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M278">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M279">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M281">View MathML</a>. Consequently, by Theorem 2.7, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M243">View MathML</a> has a fixed point u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M283">View MathML</a> which is a solution to the problem (6)-(8). □

6 An example

Consider the following neutral integro-differential equations of fractional order:

(16)

(17)

(18)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M287">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M288">View MathML</a>

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M289">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M290">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M290">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M293">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M294">View MathML</a>

Hence, the condition (H1) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M297">View MathML</a>. Also, the condition (H2) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M299">View MathML</a>.

We shall show that the condition (9) holds for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M300">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M301">View MathML</a>. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M303">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M304">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M305">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M306">View MathML</a>

Consequently, Theorem 3.2 implies that the problem (16)-(18) has at least one solution defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/128/mathml/M307">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The three authors have participated into the obtained results. The collaboration of each one cannot be separated in different parts of the paper. All of them have made substantial contributions to the theoretical results. The three authors have been involved in drafting the manuscript and revising it critically for important intellectual content. All authors have given final approval of the version to be published.

Acknowledgements

The authors are grateful to the referees for their helpful remarks. Third author is partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.

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