This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

A new approach to BVPs with state-dependent impulses

Irena Rachůnková* and Jan Tomeček

Author Affiliations

Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

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Boundary Value Problems 2013, 2013:22  doi:10.1186/1687-2770-2013-22


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/22


Received:24 October 2012
Accepted:15 January 2013
Published:11 February 2013

© 2013 Rachůnková and Tomeček; 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

The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M1">View MathML</a>

Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2">View MathML</a>. Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions.

MSC: 34B37, 34B15.

Keywords:
impulsive differential equation; state-dependent impulses; Dirichlet problem; second-order ODE

1 Introduction

Differential equations involving impulse effects appear as a natural description of observed evolution phenomena of several real world problems. We refer to the monographs [1-3].

Most papers in the literature on impulsive boundary value problems concern the case with fixed moments of impulsive effects. Papers dealing with state-dependent impulses, called also impulses at variable times, focus their attention on initial value problems or periodic problems. Such papers investigate the existence, stability or asymptotic properties of solutions of initial value problems [4-8] or solvability of autonomous periodic problems [9,10] and nonautonomous ones [11-15]. We can also find papers investigating other boundary value problems with state-dependent impulses through some initial value problems for multi-valued maps [16,17].

In this paper we provide a new approach to boundary value problems with state-dependent impulses based on a construction of proper sets and operators and the topological degree arguments. Unlike previous existing results, our approach enables us to find simple existence conditions for data functions and it can be used for other regular (and also singular) problems. We demonstrate it on the second-order Dirichlet boundary value problem with one state-dependent impulse

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M3">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M4">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M5">View MathML</a>

(3)

where we assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M6">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M7">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M8">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M9">View MathML</a>

(7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M10">View MathML</a>

(8)

Under assumptions (4)-(8), we prove the solvability of problem (1)-(3). In particular, we transform problem (1)-(3) to a fixed point problem for a proper operator in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2">View MathML</a>. This approach can be also used for other types of boundary conditions and it can be easily extended to more impulses.

Here, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M12">View MathML</a> the set of all continuous functions on the interval J, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M13">View MathML</a> the set of all functions having continuous derivatives on the interval J and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M14">View MathML</a> the set of all Lebesgue integrable functions on J. For a compact interval J, we consider the linear space of functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M12">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M13">View MathML</a> equipped, respectively, with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M17">View MathML</a>

In this paper we work with the linear space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M19">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M20">View MathML</a>

It is well-known that the mentioned normed spaces are Banach spaces. Recall that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M21">View MathML</a>, a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M22">View MathML</a> satisfies the Carathéodory conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M23">View MathML</a> (we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M24">View MathML</a>) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M25">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M26">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M27">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M28">View MathML</a>,

• for each compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M29">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M31">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M28">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M33">View MathML</a>.

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M34">View MathML</a> is a solution of problem (1)-(3), if z is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35">View MathML</a>, there exists unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M36">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M39">View MathML</a> have absolutely continuous first derivatives, z satisfies equation (1) for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a> and fulfills conditions (2), (3).

2 Operators

In this section we assume that (4)-(8) are fulfilled. We introduce sets and operators corresponding to problem (1)-(3) and prove their properties which are needed for an application of the Leray-Schauder degree theory. Let us consider K of (7) and define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M41">View MathML</a>

Lemma 1For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42">View MathML</a>, there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M43">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M44">View MathML</a>

(9)

Proof Let us take an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42">View MathML</a>. Obviously, the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M47">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46">View MathML</a> is a root of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M49">View MathML</a>

From (8) it follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M51">View MathML</a>. According to (8) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M52">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M53">View MathML</a>

(10)

Therefore, σ is strictly decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35">View MathML</a> and hence it has exactly one root in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M55">View MathML</a>. □

Now, define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M56">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M57">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46">View MathML</a> fulfills (9). The next lemma provides an important result about the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M59">View MathML</a> which is fundamental for our approach.

Lemma 2The functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M59">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M61">View MathML</a>.

Proof Let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M42">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M65">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66">View MathML</a>. Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M67">View MathML</a>

By Lemma 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M69">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M71">View MathML</a>, respectively. According to (8), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M73">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M74">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M75">View MathML</a>

(11)

We will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M76">View MathML</a>. Let us take an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M77">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M79">View MathML</a> (cf. (10)), we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M81">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M82">View MathML</a>

From (11) it follows the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M83">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M84">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M85">View MathML</a>. By Lemma 1 and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M72">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M87">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M88">View MathML</a>. □

Further, consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M89">View MathML</a> of (8) and define sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M90">View MathML</a> and Ω by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M92">View MathML</a>

(12)

Finally, define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M93">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M94">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M95">View MathML</a>

(13)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M97">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M98">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M99">View MathML</a>

(15)

and G is the Green function of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M101">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M102">View MathML</a>

Lemma 3The operatoris compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M103">View MathML</a>.

Proof First, we will prove the continuity of the operator ℱ. Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M107">View MathML</a>

(16)

Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M111">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64">View MathML</a>. We will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M113">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>, we get by (13)-(15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M116">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M117">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M118">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M119">View MathML</a>

By (16), there exists a compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M120">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M122">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M124">View MathML</a>. Consequently, by (4), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M126">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M64">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M129">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127">View MathML</a>, then due to the Lebesgue dominated convergence theorem, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M131">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M132">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M76">View MathML</a>, the absolute continuity of the Lebesgue integral yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M134">View MathML</a>

Further, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M135">View MathML</a> the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M136">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127">View MathML</a> and the same is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M138">View MathML</a>. The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M138">View MathML</a> and I imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M141">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M132">View MathML</a> uniformly w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M144">View MathML</a> converges to x in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M66">View MathML</a>. Similar arguments can be applied to the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M146">View MathML</a>.

Now we will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147">View MathML</a> is relatively compact. The boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M103">View MathML</a> implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M150">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M152">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M153">View MathML</a>

Therefore, by (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M154">View MathML</a>

We have proved that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M156">View MathML</a>. We now show that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M157">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M35">View MathML</a>. For a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M159">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M160">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M161">View MathML</a>

As a result, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M77">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M163">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M164">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M165">View MathML</a>, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M166">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M167">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M147">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M2">View MathML</a> by the Arzelà-Ascoli theorem. □

Lemma 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a>be a fixed point of ℱ. Then the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M171">View MathML</a>

(17)

is a solution of problem (1)-(3).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M173">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M174">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M97">View MathML</a>.

Let us consider the function z defined in (17). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M178">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M179">View MathML</a>

(19)

and by Lemma 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M180">View MathML</a>

(20)

In addition, by (17), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46">View MathML</a> is a unique point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M182">View MathML</a> satisfying (20). Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M184">View MathML</a>. Due to (19) and (20), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M185">View MathML</a>. Further,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M186">View MathML</a>

Therefore, σ is strictly decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M187">View MathML</a>, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M189">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M46">View MathML</a> is a unique point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M55">View MathML</a> satisfying (20).

Further, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M192">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>, and by virtue of (14),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M194">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M195">View MathML</a>

Finally,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M196">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M197">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M198">View MathML</a>

 □

3 Main result

Here, using the Leray-Schauder degree theory, we prove our main result about the solvability of problem (1)-(3). To this end, we will need the following lemma on a priori estimates.

Lemma 5Assume (4)-(8). Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M199">View MathML</a>and any solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M200">View MathML</a>of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M201">View MathML</a>

(21)

the implication

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M202">View MathML</a>

(22)

holds.

Proof Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M203">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a> satisfy (21), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M205">View MathML</a>

(23)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M40">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M207">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M105">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M210">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M214">View MathML</a>. There are two possibilities as follows.

Case A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M215">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M216">View MathML</a> and from (15) and (24), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M217">View MathML</a>

which implies, due to (6) and (8),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M218">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M219">View MathML</a>, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M220">View MathML</a>.

Case B. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M221">View MathML</a>. From (24), (14), (6), (5) and (8), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M222">View MathML</a>

This inequality together with (7) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M223">View MathML</a>

which is a contradiction.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M224">View MathML</a>, the solution of (21) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M225">View MathML</a>, and it clearly belongs to Ω. □

Theorem 6Assume (4)-(8). Then the operatorhas a fixed point in Ω.

Proof According to Lemma 5, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M226">View MathML</a> is a homotopy. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M227">View MathML</a>

and consequently the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M228">View MathML</a>

has a solution in Ω. This solution is a fixed point of the operator ℱ. □

Theorem 7Assume (4)-(8). Then problem (1)-(3) has a solutionzsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M229">View MathML</a>

(25)

Proof From Theorem 6 it follows that there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M220">View MathML</a> of the operator ℱ. Lemma 4 yields that the function z defined in (17) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M231">View MathML</a>) is a solution of problem (1)-(3). Estimates (25) follow from (17) and from the definitions of Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M89">View MathML</a> (cf. (12) and (8)). □

Remark 8 Let us note that assumption (7) follows from the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M233">View MathML</a>

4 Examples

In this section we demonstrate that Theorem 7 can be applied to sublinear, linear and superlinear problems.

Example 9 (Sublinear problem)

Let us consider problem (1)-(3) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M234">View MathML</a>

that is, f and I are sublinear in x. Then assumptions (5) and (6) are valid for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M235">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M236">View MathML</a>

Remark 8 yields that condition (7) is satisfied for any sufficiently large K. In particular, let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M237">View MathML</a>

If we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M238">View MathML</a>, we see that (7) holds. Then by (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M239">View MathML</a>

For instance, if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M240">View MathML</a> and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M241">View MathML</a>

(26)

or if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M243">View MathML</a> and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M244">View MathML</a>

(27)

we can check that conditions (8) are satisfied in both cases. Therefore, by Theorem 7, the corresponding problem (1)-(3) has at least one solution.

Note that (27) shows that γ need not be monotonous.

Example 10 (Linear problem)

Let us consider problem (1)-(3) with f and I having the linear behavior in x and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M245">View MathML</a>

Then assumptions (5) and (6) are valid for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M246">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M247">View MathML</a>

Theorem 7 can be applied, due to Remark 8, under the additional assumption

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M248">View MathML</a>

(28)

If (28) holds, then for any sufficiently large K, condition (7) is satisfied. By (8), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M249">View MathML</a>, and problem (1)-(3) has a solution for any γ satisfying (8). Consequently, if γ is given by (26) or (27), problem (1)-(3) is solvable.

Example 11 (Superlinear problem) Let us consider problem (1)-(3) with f and I superlinear in x. Put, for example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M250">View MathML</a>

(29)

Then assumptions (5) and (6) are valid for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M251">View MathML</a>

It holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M252">View MathML</a>

By virtue of (7), Theorem 7 can be applied provided there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M253">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M254">View MathML</a>

(30)

Let us search K in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M255">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M256">View MathML</a> and it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M257">View MathML</a>

Consequently, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M258">View MathML</a> fulfilling the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M259">View MathML</a>

satisfies (30) as well. Put, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M261">View MathML</a>. Then we get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M262">View MathML</a> inequality (30) holds. Consequently, (8) gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/22/mathml/M263">View MathML</a> and the corresponding problem (1)-(3) is solvable for any γ satisfying (8). In particular, γ given by (26) or (27) can be considered in this case as well.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

Dedicated to Jean Mawhin on the occasion of his 70th birthday.

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by the grant Matematické modely a struktury, PrF_ 2012_ 017.

References

  1. Bainov, D, Simeonov, P: Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Essex (1993)

  2. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

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

  4. Afonso, SM, Bonotto, EM, Federson, M, Schwabik, Š: Discontinuous local semiflows for Kurzweil equations leading to LaSalle’s invariance principle for differential systems with impulses at variable times. J. Differ. Equ.. 250, 2969–3001 (2011). Publisher Full Text OpenURL

  5. Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: Impulsive functional differential equations with variable times. Comput. Math. Appl.. 47, 1659–1665 (2004). Publisher Full Text OpenURL

  6. Domoshnitsky, A, Drakhlin, M, Litsyn, E: Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments. J. Differ. Equ.. 228, 39–48 (2006). Publisher Full Text OpenURL

  7. Frigon, M, O’Regan, D: Impulsive differential equations with variable times. Nonlinear Anal.. 26, 1913–1922 (1996). Publisher Full Text OpenURL

  8. Kaul, S, Lakshmikantham, V, Leela, S: Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times. Nonlinear Anal.. 22, 1263–1270 (1994). Publisher Full Text OpenURL

  9. Liu, L, Sun, J: Existence of periodic solution of a harvested system with impulses at variable times. Phys. Lett. A. 360, 105–108 (2006). Publisher Full Text OpenURL

  10. Qi, J, Fu, X: Existence of limit cycles of impulsive differential equations with impulses at variable times. Nonlinear Anal.. 44, 345–353 (2001). Publisher Full Text OpenURL

  11. Bajo, I, Liz, E: Periodic boundary value problem for first order differential equations with impulses at variable times. J. Math. Anal. Appl.. 204, 65–73 (1996). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  12. Belley, J, Virgilio, M: Periodic Duffing delay equations with state dependent impulses. J. Math. Anal. Appl.. 306, 646–662 (2005). Publisher Full Text OpenURL

  13. Belley, J, Virgilio, M: Periodic Liénard-type delay equations with state-dependent impulses. Nonlinear Anal.. 64, 568–589 (2006). Publisher Full Text OpenURL

  14. Frigon, M, O’Regan, D: First order impulsive initial and periodic problems with variable moments. J. Math. Anal. Appl.. 233, 730–739 (1999). Publisher Full Text OpenURL

  15. Yong, L, Fuzhong, C, Zhanghua, L: Boundary value problems for impulsive differential equations. Nonlinear Anal. TMA. 29, 1253–1264 (1997). Publisher Full Text OpenURL

  16. Benchohra, M, Graef, JR, Ntouyas, SK, Ouahab, A: Upper and lower solutions method for impulsive differential inclusions with nonlinear boundary conditions and variable times. Dyn. Contin. Discrete Impuls. Syst.. 12, 383–396 (2005)

  17. Frigon, M, O’Regan, D: Second order Sturm-Liouville BVP’s with impulses at variable times. Dyn. Contin. Discrete Impuls. Syst.. 8, 149–159 (2001)