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Positive solutions for a sixth-order boundary value problem with four parameters

Ravi P Agarwal1*, B Kovacs2 and D O’Regan3

Author Affiliations

1 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, 78363-8202, USA

2 Department of Analysis, University of Miskolc, Egyetemvaros, 3515, Hungary

3 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

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Boundary Value Problems 2013, 2013:184  doi:10.1186/1687-2770-2013-184


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/184


Received:20 May 2013
Accepted:30 July 2013
Published:14 August 2013

© 2013 Agarwal 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

This paper investigates the existence and multiplicity of positive solutions of a sixth-order differential system with four variable parameters using a monotone iterative technique and an operator spectral theorem.

MSC: 34B15, 34B18.

Keywords:
positive solutions; variable parameters; fixed point theorem; operator spectral theorem

1 Introduction

It is well known that boundary value problems for ordinary differential equations can be used to describe a large number of physical, biological and chemical phenomena. In recent years, boundary value problems for sixth-order ordinary differential equations, which arise naturally, for example, in sandwich beam deflection under transverse shear have been studied extensively, see [1-4] and the references therein. The deformation of the equilibrium state of an elastic circular ring segment with its two ends simply supported can be described by a boundary value problem involving a sixth-order ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M1">View MathML</a>

(1)

Liu and Li [5] studied the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M2">View MathML</a>

(2)

They showed that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3">View MathML</a> such that the above boundary value problem has at least two, one, and no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M6">View MathML</a>, respectively.

In this paper, we discuss the existence of positive solutions for the sixth-order boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M7">View MathML</a>

(3)

For this, we shall assume the following conditions throughout

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M8">View MathML</a> is continuous;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M12">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M16">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M19">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M21">View MathML</a>.

Assumption (H2) involves a three-parameter nonresonance condition.

More recently Li [6] studied the existence and multiplicity of positive solutions for a sixth-order boundary value problem with three variable coefficients. The main difference between our work and [6] is that we consider boundary value problem not only with three variable coefficients, but also with two positive parameters λ and μ, and the existence of the positive solution depends on these parameters. In this paper, we shall apply the monotone iterative technique [7] to boundary value problem (3) and then obtain several new existence and multiplicity results. In the special case, in [8] by using the fixed point theorem and the operator spectral theorem, we establish a theorem on the existence of positive solutions for the sixth-order boundary value problem (3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22">View MathML</a>.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M24">View MathML</a>. It is well known that Y is a Banach space equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M25">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M26">View MathML</a>. For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M28">View MathML</a>, we denote the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M29">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M30">View MathML</a>

We also need the space X, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M31">View MathML</a>

In [8], it is shown that X is complete with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M33">View MathML</a>, and moreover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M35">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M36">View MathML</a>, consider the linear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M37">View MathML</a>

(4)

where a, b, c satisfy the assumption

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M38">View MathML</a>

(5)

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M39">View MathML</a>. Inequality (5) follows immediately from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M40">View MathML</a> is the first eigenvalue of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M42">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M43">View MathML</a> is the first eigenfunction, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M44">View MathML</a>. Since the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M45">View MathML</a> is the first eigenvalue line of the three-parameter boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M47">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M48">View MathML</a> lies in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M49">View MathML</a>, then by the Fredholm alternative, the existence of a solution of the boundary value problem (4) cannot be guaranteed.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M50">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M52">View MathML</a>. It is easy to see that the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M53">View MathML</a> has two real roots <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M54">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M55">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M56">View MathML</a> be a number such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M57">View MathML</a>. In this case, (4) satisfies the decomposition form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M58">View MathML</a>

(6)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60">View MathML</a>) is the Green’s function associated with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M61">View MathML</a>

(7)

We need the following lemmas.

Lemma 1[5,9]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M62">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60">View MathML</a>) can be expressed as

(i) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M65">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M66">View MathML</a>

(ii) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M67">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M68">View MathML</a>

(iii) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M69">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M70">View MathML</a>

Lemma 2[5]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M60">View MathML</a>) has the following properties

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M74">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M76">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M78">View MathML</a>,

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M80">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M65">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M83">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M67">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M86">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M87">View MathML</a>.

In what follows, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M88">View MathML</a>.

Lemma 3[10]

LetXbe a Banach space, Ka cone and Ω a bounded open subset ofX. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M90">View MathML</a>be condensing. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M91">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M93">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M94">View MathML</a>.

Lemma 4[10]

LetXbe a Banach space, letKbe a cone ofX. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M95">View MathML</a> (here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M97">View MathML</a>) is a compact map such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M98">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M99">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M100">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M99">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M102">View MathML</a>.

Now, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M103">View MathML</a>

(8)

the solution of boundary value problem (4) can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M104">View MathML</a>

(9)

Thus, for every given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M36">View MathML</a>, the boundary value problem (4) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M106">View MathML</a>, which is given by (9).

We now define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M107">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M108">View MathML</a>

(10)

Throughout this article, we shall denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M109">View MathML</a> the unique solution of the linear boundary value problem (4).

Lemma 5[8]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M110">View MathML</a>is linear completely continuous, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M113">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M115">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M116">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M118">View MathML</a>.

We list the following conditions for convenience

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119">View MathML</a> is nondecreasing in u for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M120">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M121">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M123">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>;

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M125">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M128">View MathML</a> is independent of ρ and u.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M129">View MathML</a> is the Green’s function of the linear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M130">View MathML</a>

(11)

Then, the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M131">View MathML</a>

can be solved by using Green’s function, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M132">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M133">View MathML</a>. Thus, inserting (12) into the first equation in (3), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M134">View MathML</a>

(13)

Let us consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M135">View MathML</a>

(14)

Now, we consider the existence of a positive solution of (14). The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M136">View MathML</a> is a positive solution of (14), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M139">View MathML</a>.

Let us rewrite equation (13) in the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M140">View MathML</a>

(15)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M141">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M142">View MathML</a>

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M143">View MathML</a> is linear. By Lemmas 2 and 3 in [8], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M146">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M151">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M152">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M153">View MathML</a>. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M154">View MathML</a> is a solution of (13) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M141">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M156">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M157">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M158">View MathML</a>

(16)

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M159">View MathML</a> maps X into X. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M113">View MathML</a> together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M161">View MathML</a> and the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M162">View MathML</a>, and applying the operator spectral theorem, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M163">View MathML</a> exists and is bounded. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M164">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M165">View MathML</a>, then the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M166">View MathML</a> is fulfilled. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M167">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M168">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M169">View MathML</a>. Then (16) is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M170">View MathML</a>. By the Neumann expansion formula, H can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M171">View MathML</a>

(17)

The complete continuity of T with the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M172">View MathML</a> guarantees that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M173">View MathML</a> is completely continuous.

Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M115">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M116">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M118">View MathML</a>. Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M179">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M180">View MathML</a>

(18)

and so, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>.

It is easy to see [11] that the following inequalities hold: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M184">View MathML</a>

(19)

and, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M185">View MathML</a>

(20)

Lemma 6[8]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M186">View MathML</a>is completely continuous, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, and, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M192">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M193">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M194">View MathML</a>. From (H1), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M195">View MathML</a> is continuous. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M196">View MathML</a>, being a positive solution of (13), is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M193">View MathML</a>, being a nonzero solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M198">View MathML</a>

(21)

Let us introduce the following notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M199">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M200">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M201">View MathML</a> is completely continuous. We next show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> has a nonzero fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M203">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M204">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M205">View MathML</a>. It is easy to see that P is a cone in Y, and now, we show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M206">View MathML</a>.

Lemma 7<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M207">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M208">View MathML</a>is completely continuous.

Proof It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M209">View MathML</a> is completely continuous. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M210">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M211">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M212">View MathML</a>. Using Lemma 6, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a> and by Lemma 2, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M215">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M216">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M217">View MathML</a>

(22)

On the other hand, by Lemma 6 and (22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M218">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M207">View MathML</a>. □

3 Main results

Lemma 8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119">View MathML</a>be nondecreasing inufor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M221">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M222">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M224">View MathML</a>is a constant and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M227">View MathML</a>such that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a>has a fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M230">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M231">View MathML</a>.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M232">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M233">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M234">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M235">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M236">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M238">View MathML</a>, respectively. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M240">View MathML</a>, and from Lemma 6, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M241">View MathML</a>

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M242">View MathML</a>

Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M243">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M244">View MathML</a>, then from (10), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M245">View MathML</a>. Using equation (4) and (6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M246">View MathML</a>

(23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M247">View MathML</a>

(24)

Then by (23), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M248">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M249">View MathML</a>

because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M250">View MathML</a>, and finally, from (24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M251">View MathML</a>

because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M253">View MathML</a>. From the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M254">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M255">View MathML</a>

(25)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M256">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>. Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M259">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M260">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M261">View MathML</a>

(26)

and from (26), it follows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M262">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M263">View MathML</a> then, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M264">View MathML</a>

(27)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M269">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M270">View MathML</a>

 □

Indeed, by Lemma 6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M271">View MathML</a>

Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119">View MathML</a> nondecreasing in u for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, Lemma 2, and the Lebesgue convergence theorem guarantee that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M274">View MathML</a> decreases to a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M231">View MathML</a> of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M276">View MathML</a>.

Lemma 9Suppose that (H3)-(H5) hold, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M278">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M279">View MathML</a>for some constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M281">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M282">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M283">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M284">View MathML</a>.

Proof Suppose, to the contrary, that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M285">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M286">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M287">View MathML</a> is a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M289">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267">View MathML</a>). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M291">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M292">View MathML</a>.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M293">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M294">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M296">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M297">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M298">View MathML</a>. Now,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M299">View MathML</a>

and so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M300">View MathML</a>

which is a contradiction. □

Lemma 10Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>, (H3) and (H4) hold and that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a>has a positive fixed point inPat<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M303">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M304">View MathML</a>. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M305">View MathML</a>there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M307">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M308">View MathML</a> be a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M310">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M311">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M313">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M314">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M315">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M316">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M317">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M318">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M319">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M320">View MathML</a>

because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M119">View MathML</a> is nondecreasing in u for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M323">View MathML</a> is also nondecreasing in u. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M324">View MathML</a>

(28)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M325">View MathML</a>

 □

Indeed, by Lemma 6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M326">View MathML</a>

Lemma 2 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M327">View MathML</a> decreases to a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306">View MathML</a>.

Lemma 11Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>, (H3)-(H5) hold. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M330">View MathML</a>

Then Λ is bounded.

Proof Suppose, to the contrary, that there exists a fixed point sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M331">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M332">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M333">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M335">View MathML</a>. Then there are two cases to be considered: (i) there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M336">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M337">View MathML</a>, which is impossible by Lemma 9, so we only consider the next case: (ii) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M338">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M340">View MathML</a> . In view of (H3) and (H4), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M341">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M342">View MathML</a>, and further, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M343">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>. We know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M345">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M346">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M347">View MathML</a>. Then it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M348">View MathML</a>

(29)

Multiplying (29) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M349">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M350">View MathML</a>, and then using integration by parts on the left side of (29), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M351">View MathML</a>

Next, assume that (ii) holds. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M352">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M353">View MathML</a>

lead to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M354">View MathML</a>, which is a contradiction. The proof is complete. □

Lemma 12Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>, (H3)-(H4) hold. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M356">View MathML</a>

and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M357">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M358">View MathML</a>, where Λ is defined in Lemma 11.

Proof By Lemma 10, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M359">View MathML</a>. We only need to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M360">View MathML</a>. We may choose a distinct nondecreasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M361">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M362">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M287">View MathML</a> as a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267">View MathML</a> , i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M367">View MathML</a>. By Lemma 9, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M368">View MathML</a> is uniformly bounded, so it has a subsequence, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M369">View MathML</a>, converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M370">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M371">View MathML</a>

(30)

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M372">View MathML</a> on both sides of (30), and using the Lebesgue convergence theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M373">View MathML</a>

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> has a positive fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M375">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M376">View MathML</a>. □

Theorem 1Suppose that (H3)-(H5) hold, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>. For fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M378">View MathML</a>, then there exists at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3">View MathML</a>such that (3) has at least two, one and has no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M381">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M382">View MathML</a>, respectively.

Proof Suppose that (H3) and (H4) hold. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M383">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M384">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M386">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M388">View MathML</a>. In view of Lemma 12, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M202">View MathML</a> also has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M390">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M391">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M392">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M393">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M394">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M395">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M396">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M397">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M399">View MathML</a>. In this case, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M400">View MathML</a>

Indeed, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M401">View MathML</a>

Similarly, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M402">View MathML</a>

Moreover, from (25), it follows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M403">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M404">View MathML</a>

Finally, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M405">View MathML</a>

By induction, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M406">View MathML</a>

(31)

Hence, using (31), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M407">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M408">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M409">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M410">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M411">View MathML</a> is completely continuous. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M412">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M414">View MathML</a>. Indeed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M414">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M416">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M417">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M418">View MathML</a>

By Lemma 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M419">View MathML</a>.

Let k be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M420">View MathML</a>

We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M421">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, so we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M423">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M424">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M425">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M426">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M427">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M428">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M429">View MathML</a> is completely continuous. It is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M430">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M432">View MathML</a>. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M433">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M434">View MathML</a>

In view of Lemma 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M435">View MathML</a>. By the additivity of the fixed point index,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M436">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M437">View MathML</a> has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M438">View MathML</a> and has another fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M439">View MathML</a> by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M440">View MathML</a>. □

Let us introduce the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441">View MathML</a> in the equation of (13), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M442">View MathML</a>

(32)

In this case, we can prove the following theorem, which is similar to Theorem 1.

Theorem 2Suppose that (H3)-(H5) hold, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>. Then there exists at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M3">View MathML</a>such that (32) has at least two, one and has no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M446">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M382">View MathML</a>, respectively.

We follow exactly the same procedure, described in detail in the proof of Theorem 1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441">View MathML</a>.

Let us introduce the following notations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M451">View MathML</a>

(33)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M452">View MathML</a>.

Lemma 13Suppose that (H3), (H4) and (H6) hold, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M454">View MathML</a>, there exist real numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M455">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M456">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M457">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M458">View MathML</a> from Lemma 6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M459">View MathML</a>

Note that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M458">View MathML</a>, there exists an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M461">View MathML</a> and a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M462">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M463">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M464">View MathML</a>. In addition, by (H6), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M465">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M466">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M467">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M464">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M469">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M470">View MathML</a>; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M471">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M472">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M473">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M474">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M475">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M476">View MathML</a>. □

Theorem 3Suppose that (H3), (H4) and (H6) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22">View MathML</a>. Then

(i) (32) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M479">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M480">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M481">View MathML</a>are constants.

(ii) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M482">View MathML</a>, the sequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M483">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267">View MathML</a>) converges uniformly to the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229">View MathML</a>, and the rate of convergence is determined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M486">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M487">View MathML</a>is a positive number.

Proof In view of (H3), (H4) and (H6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M488">View MathML</a> is a nondecreasing operator and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M489">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M491">View MathML</a>. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M492">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M493">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M494">View MathML</a> is nondecreasing in u, then by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M495">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M497">View MathML</a>

Moreover, from (25), it follows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M498">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M499">View MathML</a>

(34)

Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M494">View MathML</a> is nondecreasing in u, then by using form (34), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M501">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M503">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M504">View MathML</a>

By induction, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M505">View MathML</a>

(35)

Hence, using (35), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M506">View MathML</a>

(36)

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M507">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M489">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M510">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M511">View MathML</a>

Moreover, from (25), it follows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M512">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M513">View MathML</a>

Finally, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M514">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M515">View MathML</a>

By induction, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M516">View MathML</a>

(37)

Hence, using (35) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M517">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M518">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M519">View MathML</a>

(38)

By Lemma 13, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M520">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M521">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M522">View MathML</a>

Pick <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M523">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M524">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M525">View MathML</a>

(39)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M526">View MathML</a>

(40)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M527">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M528">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M529">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M530">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M267">View MathML</a> . From (36) and (38), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M532">View MathML</a>

(41)

Indeed, from (39) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M533">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M534">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M535">View MathML</a>

and by induction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M536">View MathML</a>

From (40), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M537">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M538">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M539">View MathML</a>

and by induction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M540">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M541">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M542">View MathML</a>

(42)

In fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M543">View MathML</a> is clear. Assume that (42) holds with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M544">View MathML</a> (k is a positive integer), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M545">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M546">View MathML</a>

By induction, it is easy to see that (42) holds. Furthermore, in view of (38), (41) and (42), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M547">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M548">View MathML</a>

where z is a nonnegative integer. Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M549">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M550">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M551">View MathML</a> is a fixed point of Q and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M552">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M553">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M554">View MathML</a>.

Next we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M229">View MathML</a> is the unique fixed point of Q in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M556">View MathML</a>. Suppose, to the contrary, that there exists another <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M557">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M558">View MathML</a>. We can suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M559">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M560">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M561">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M562">View MathML</a>. We assert <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M563">View MathML</a>. Otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M564">View MathML</a>, and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M565">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M566">View MathML</a>, which is a contradiction of the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M567">View MathML</a>, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M568">View MathML</a>.

Let us introduce the following notations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M441">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M570">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M571">View MathML</a>, where Q is given by (33). □

Theorem 4Suppose that (H3), (H4), (H6) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M225">View MathML</a>hold. Then (32) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M573">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M574">View MathML</a>.

Proof Theorem 3 implies that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M22">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M576">View MathML</a> has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M577">View MathML</a>, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M578">View MathML</a>. Then from Lemma 10, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M579">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M306">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M581">View MathML</a>.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M582">View MathML</a> is a unique positive solution of (32) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M574">View MathML</a>. □

4 Application

As an application of Theorem 1, consider the sixth-order boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M584">View MathML</a>

(43)

for a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M587">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M588">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M589">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M590">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M591">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M592">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M593">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M594">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M595">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M596">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M597">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M598">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M599">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M600">View MathML</a>. Note also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M601">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M602">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M603">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M604">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M605">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M606">View MathML</a>. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M607">View MathML</a>, then the conditions of Theorem 1 (note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/184/mathml/M608">View MathML</a>) are fulfilled (in particular, (H3)-(H5) are satisfied). As a result, Theorem 1 can be applied to (43).

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