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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Highly Accessed Research

Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance

Juan J Nieto

Author Affiliations

Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Boundary Value Problems 2013, 2013:130  doi:10.1186/1687-2770-2013-130

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/130


Received:20 February 2013
Accepted:30 April 2013
Published:20 May 2013

© 2013 Nieto; 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

We present a new existence result for a second-order nonlinear ordinary differential equation with a three-point boundary value problem when the linear part is noninvertible.

MSC: 34B10, 34B15.

Keywords:
nonlinear ordinary differential equation; three-point boundary value problem; problem at resonance; existence of solution

1 Introduction

The study of multi-point boundary value problems for linear second-order ordinary differential equations goes back to the method of separation of variables [1]. Also, some questions in the theory of elastic stability are related to multi-point problems [2]. In 1987, Il’in and Moiseev [3,4] studied some nonlocal boundary value problems. Then, for example, Gupta [5] considered a three-point nonlinear boundary value problem. For some recent works on nonlocal boundary value problems, we refer, for example, to [6-15] and references therein.

As indicated in [16], there has been enormous interest in nonlinear perturbations of linear equations at resonance since the seminal paper of Landesman and Lazer [17]; see [18] for further details.

Here we study the following nonlinear ordinary differential equation of second order subject to the three-point boundary condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M3">View MathML</a> is a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M5">View MathML</a>.

In this paper we consider the resonance case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M6">View MathML</a> to obtain a new existence result. Although this situation has already been considered in the literature [19], we point out that our approach and methodology is different.

2 Linear problem

Consider the linear second-order three-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M7">View MathML</a>

(2)

for a given function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M8">View MathML</a>.

The general solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M9">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M11">View MathML</a> arbitrary constants.

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M12">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M13">View MathML</a>. From the second boundary condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M14">View MathML</a>

(3)

2.1 Nonresonance case

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M15">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M16">View MathML</a>

and the linear problem (2) has a unique solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M17">View MathML</a>. In this case, we say that (2) is a nonresonant problem since the homogeneous problem has only the trivial solution as a solution, i.e., when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M20">View MathML</a>. Note that the solution is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M21">View MathML</a>

(4)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M22">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M23">View MathML</a> this is precisely the function given in Lemma 2.3 of [20] or in Remark 12 of [21].

2.2 Resonance case

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M24">View MathML</a>, then (3) is solvable if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M25">View MathML</a>

(5)

and then (2) has a solution if and only if (5) holds. In such a case, (2) has an infinite number of solutions given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M26">View MathML</a>

In particular ct, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27">View MathML</a> is a solution of the homogeneous linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M28">View MathML</a>

satisfying the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M29">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M30">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M31">View MathML</a>

We now use that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M32">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M33">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M34">View MathML</a>

Hence the solution of (2) is given, implicitly, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M35">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M36">View MathML</a>

(6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M37">View MathML</a>

We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M39">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M40">View MathML</a>.

3 Nonlinear problem

Defining the operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M41">View MathML</a>

the nonlinear problem is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M42">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M43">View MathML</a>.

We note that (6) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M44">View MathML</a>

and the nonlinear problem (1) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M45">View MathML</a>

This suggests to introduce the new function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M46">View MathML</a>. To find a solution u, we have to find v and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M47">View MathML</a>.

For every constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27">View MathML</a>, we solve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M49">View MathML</a>

(7)

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M50">View MathML</a> be the set of solutions of (7). This set may be empty (no solution), a singleton (unique solution) or with more than one element (multiple solutions). For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M51">View MathML</a>, we consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M52">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M53">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M54">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M55">View MathML</a> is a solution of the nonlinear problem (1). We then look for fixed points of the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M56">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27">View MathML</a> fixed, we try to solve the integral equation (7).

Assume that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M59">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M60">View MathML</a>

(8)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M62">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M63">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M64">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M65">View MathML</a>

Thus, a solution of (7) is precisely a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M66">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M67">View MathML</a> is a compact operator. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M63">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M69">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M70">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M71">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M73">View MathML</a>

Hence there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M75">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M76">View MathML</a>

(9)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M70">View MathML</a> solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M71">View MathML</a>. This implies that v is bounded independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M80">View MathML</a>, and hence by Schaefer’s fixed point theorem (Theorem 4.3.2 of [22]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M67">View MathML</a> has at least a fixed point, i.e., for given c, equation (7) is solvable.

Now suppose f is Lipschitz continuous.

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M82">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M83">View MathML</a>

(10)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M85">View MathML</a>.

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M86">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M87">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M88">View MathML</a>

Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M82">View MathML</a> small, equation (7) has a unique solution in view of the classical Banach contraction fixed point theorem.

Now, under conditions (8) and (10), set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M90">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M91">View MathML</a> is the unique solution of (7), and as a consequence of the contraction principle, this map is continuous.

Define the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M92">View MathML</a>

If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M94">View MathML</a>, then for that c we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M95">View MathML</a>, and the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M96">View MathML</a>

is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M97">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M55">View MathML</a> is a solution of the original nonlinear problem (1).

Now, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M99">View MathML</a>

(11)

uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M61">View MathML</a>.

Then the growth of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M101">View MathML</a> is sublinear in view of estimate (9). However, c growths linearly. Hence the norm of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M102">View MathML</a>

growths asymptotically as c.

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M103">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M27">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M94">View MathML</a>.

We have the following result.

Theorem 3.1Suppose thatfsatisfies the growth conditions (8) and (10). If (11) holds, then (1) is solvable forlsufficiently small.

Note that condition (11) is crucial since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/130/mathml/M106">View MathML</a> and, in view of (5), the problem (1) may have no solution.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This research has been partially supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER. The author is thankful to the referees for their useful suggestions.

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