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On the solvability of a Neumann boundary value problem for the differential equation f ( t , x , x , x ) = 0

P Palamides1, P Kelevedjiev2* and N Popivanov3

Author affiliations

1 Naval Academy of Greece, Piraeus, 451 10, Greece

2 Department of Mathematics, Technical University of Sliven, Sliven, Bulgaria

3 Faculty of Mathematics and Informatics, ‘St. Kl. Ohridski’ University of Sofia, Sofia, Bulgaria

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Citation and License

Boundary Value Problems 2012, 2012:77  doi:10.1186/1687-2770-2012-77

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/77


Received:5 July 2012
Accepted:6 July 2012
Published:23 July 2012

© 2012 Palamides et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using barrier strip arguments, we investigate the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M2">View MathML</a>-solutions to the Neumann boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M5">View MathML</a>.

MSC: 34B15.

Keywords:
boundary value problem; equation unsolved with respect to the second derivative; Neumann boundary conditions; existence

1 Introduction

The purpose of this paper is to establish the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M6">View MathML</a>-solutions to the scalar Neumann boundary value problem (BVP)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M7">View MathML</a>

(N)

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M8">View MathML</a> and its first derivatives are continuous only on suitable subsets of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M9">View MathML</a>.

The literature devoted to the solvability of singular and nonsingular Neumann BVPs for second order ordinary differential equations whose main nonlinearities do not depend on the second derivative is vast. We quote here only [1-5] for results and references.

The solvability of the homogeneous Neumann problem for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M10">View MathML</a>, under appropriate conditions on f, has been studied in [6-8]. Results, concerning the existence of solutions to the homogeneous and nonhomogeneous Neumann problem for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M11">View MathML</a> can be found in [9] and [10] respectively. BVPs for the same equation with various linear boundary conditions have been studied in [9,11-13]. The results of [14] guarantee the solvability of BVPs for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M12">View MathML</a> with fully linear boundary conditions. BVPs for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M13">View MathML</a> with fully nonlinear boundary conditions have been studied in [15]. For results, which guarantee the solvability of the Dirichlet BVP for the same equation, in the scalar and in the vector cases, see [12] and [16] respectively.

Concerning the kind of the nonlinearity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M14">View MathML</a>, we note that it is assumed sublinear in [6], semilinear in [11] and linear with respect to x, p and q in [8,12]. Finally, in [9] and [17]f is a linear function with respect to q, while with respect to p, it is a quadratic function or satisfies Nagumo type growth conditions respectively.

As in [10,15,18,19], we use sign conditions to establish a priori bounds for x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M16">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17">View MathML</a> is a solution to a suitable family of BVPs similar to that in [10,19]. Using these a priori bounds and applying the topological transversality theorem from [20], we prove our main existence result.

2 Basic hypotheses

To formulate our hypotheses, we use the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M18">View MathML</a>

So, we assume that there are positive constants K, M and a sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M19">View MathML</a> such that:

H1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M20">View MathML</a>

there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M22">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M23">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M24">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M25">View MathML</a> is bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28">View MathML</a>.

H2.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M29">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31">View MathML</a> is as in H1.

H3. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M33">View MathML</a> are continuous for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31">View MathML</a> is as in H1.

3 Auxiliary lemmas

In order to obtain our main existence results, we use the constant K from the hypotheses to construct the family of BVPs

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39">View MathML</a> and prove the following three auxiliary results.

Lemma 3.1Let H1 hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17">View MathML</a>be a solution to (3.1)λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M42">View MathML</a>

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M43">View MathML</a>, problem (3.1)0 is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M44">View MathML</a>

The unique solution to this BVP satisfies the bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M45">View MathML</a>

Let now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M46">View MathML</a>. Then the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M47">View MathML</a>

is a solution to the homogeneous boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M48">View MathML</a>

The equation is equivalent to the following one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M49">View MathML</a>

Hence, by the intermediate value theorem, we obtain consecutively

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M50">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M51">View MathML</a> depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M54">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M55">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M56">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M57">View MathML</a> depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M54">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M61">View MathML</a>

(3.2)

Next, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M62">View MathML</a> achieves its maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M63">View MathML</a>. Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M64">View MathML</a> has also a maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M65">View MathML</a>. Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M66">View MathML</a>

(3.3)

Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M67">View MathML</a>, from (3.2) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M68">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M71">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M75">View MathML</a>.

In view of H1, from (3.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M76">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M77">View MathML</a>

Suppose now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M78">View MathML</a>. Then, from (3.4) and (3.5) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M79">View MathML</a>

(3.6)

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M80">View MathML</a> or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M81">View MathML</a>

(3.7)

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M82">View MathML</a>. Multiplying (3.6) and (3.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M83">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M84">View MathML</a>

respectively. Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M85">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M86">View MathML</a>. So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M87">View MathML</a>

which contradicts (3.3). Thus, we infer that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M62">View MathML</a> achieves its maximum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M89">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M90">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M91">View MathML</a> be the maximum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M92">View MathML</a> and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M93">View MathML</a>. Following the above reasoning and using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M94">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M95">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M96">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M97">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M98">View MathML</a> is a strictly increasing function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M99">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M100">View MathML</a> is a sufficiently small neighbourhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M101">View MathML</a>. So, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M102">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M103">View MathML</a> is a strictly decreasing function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M99">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M105">View MathML</a> can not be the maximum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M92">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M107">View MathML</a>, which is a contradiction. Assume next that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M108">View MathML</a>. Then similar to the above arguments lead again to a contradiction. Thus, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M109">View MathML</a>

The inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M110">View MathML</a>

can be obtained in the same manner. Consequently, the eventual solutions of (3.1)λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M111">View MathML</a> satisfy the bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M112">View MathML</a>

and the proof of the lemma is completed. □

Lemma 3.2Let H1 and H2 hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M17">View MathML</a>be a solution to (3.1)λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>. Then:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a>.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a>.

Proof (a) Suppose there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M120">View MathML</a> or a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M121">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M122">View MathML</a>

By Lemma 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M123">View MathML</a>

(3.8)

In particular, (3.8) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M125">View MathML</a>. Thus, in view of H2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M126">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M127">View MathML</a>

respectively. The obtained contradictions show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M128">View MathML</a>

and therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M129">View MathML</a>

which proves (a).

(b) By the mean value theorem, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M130">View MathML</a> there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M131">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M132">View MathML</a>

Since, in view of (a), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M133">View MathML</a>, from the last formula we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M134">View MathML</a>

which proves (b) and completes the proof of the lemma. □

Lemma 3.3Let H1, H2 and H3 hold. Then there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M135">View MathML</a>continuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M136">View MathML</a>and such that

(a) the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M137">View MathML</a>

is equivalent to BVP (3.1)λ.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M138">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M139">View MathML</a>.

Proof (a) We write the differential equation from (3.1)λ as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M140">View MathML</a>

(3.9)

and consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M141">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M142">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M144">View MathML</a>, we can use H2 to conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M145">View MathML</a>

(3.10)

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M146">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M147">View MathML</a>

(3.11)

Finally, from H3 we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M148">View MathML</a>

(3.12)

So, (3.10), (3.11) and (3.12) allow us to apply a well-known theorem to conclude that there is a unique function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M149">View MathML</a> which is continuous for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M150">View MathML</a> and such that the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M151">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M152">View MathML</a>

are equivalent. Now from Lemma 3.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M153">View MathML</a>

and Lemma 3.2 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M154">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>. Consequently, equation (3.9) is equivalent to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M157">View MathML</a>

which yields the first assertion.

(b) It follows immediately from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M158">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M159">View MathML</a>. □

4 The main result

Our main result is the following existence theorem, the proof of which is based on the lemmas of the previous sections and the Topological transversality theorem [20].

Theorem 4.1Let H1, H2 and H3 hold. Then problem (N) has at least one solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160">View MathML</a>.

Proof First, we observe that according to Lemma 3.3, the family of boundary value problems

is equivalent to the family (3.1)λ for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>. Next define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M163">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M164">View MathML</a>, and the maps

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M165">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M168">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M169">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>, is a continuous, linear, one-to-one map of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M172">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M173">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M39">View MathML</a> exists and is continuous. In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>, is a continuous and j is a completely continuous embedding. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M178">View MathML</a> is a compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M179">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M58">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>, are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M185">View MathML</a> respectively, the homotopy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M186">View MathML</a>

is compact. Besides, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M187">View MathML</a>

which coincides with BVP (3.13)λ. Thus, the fixed points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M188">View MathML</a> are solutions to (3.13)λ. But, from Lemma 3.1 and Lemma 3.2 it follows that the solutions to (3.13)λ are elements of U. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>, is a fixed point free on ∂U, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M189">View MathML</a> is an admissible map for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M114">View MathML</a>. Finally, we see that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M193">View MathML</a> is a constant map, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M194">View MathML</a>, where l is the unique solution to the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M195">View MathML</a>

From the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M196">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M197">View MathML</a> is an essential map (see, [20]). By the Topological transversality theorem (see, [20]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M198">View MathML</a> is also essential, i.e., problem (3.13)1 has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160">View MathML</a>-solution. It is also a solution to (3.1)1, by Lemma 3.3. To complete the proof, remark that problem (3.1)1 coincides with the problem (N). □

We conclude with the following example, which illustrates our main result.

Example 4.2 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M200">View MathML</a>

It is clear that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M201">View MathML</a> the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M202">View MathML</a>

is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M204">View MathML</a>. Thus H1 holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M206">View MathML</a>.

To verify H2 we choose, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M209">View MathML</a>. Next we need the constants L and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M31">View MathML</a>. Having in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M212">View MathML</a>, from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M213">View MathML</a>

it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M214">View MathML</a>. On the other hand, from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M215">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M216">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M217">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M218">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28">View MathML</a>, which means that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M28">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M221">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M222">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M223">View MathML</a>

and we see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M224">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M226">View MathML</a>

Thus, H2 also holds.

Finally, H3 holds since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M33">View MathML</a> are continuous for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M201">View MathML</a>.

Thus, we can apply Theorem 4.1 to conclude that the considered problem has a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/77/mathml/M160">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same engagement.

Acknowledgements

In memory of Professor Myron K. Grammatikopoulos, 1938-2007.

This research was partially supported by Sofia University Grant N350/2012. The research of N. Popivanov was partially supported by the Bulgarian NSF under Grants DO 02-75/2008 and DO 02-115/2008.

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