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

Open Access Research

Triple solutions of complementary Lidstone boundary value problems via fixed point theorems

Patricia JY Wong

Author Affiliations

School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

Boundary Value Problems 2014, 2014:125  doi:10.1186/1687-2770-2014-125

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/125


Received:25 June 2013
Accepted:15 August 2013
Published:20 May 2014

© 2014 Wong; licensee Springer.

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

Abstract

We consider the following complementary Lidstone boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M1">View MathML</a>

By using fixed point theorems of Leggett-Williams and Avery, we offer several criteria for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained. We note that the nonlinear term F depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2">View MathML</a> and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem.

MSC: 34B15, 34B18.

Keywords:
positive solutions; complementary Lidstone boundary value problems; derivative-dependent nonlinearity; fixed point theorems

1 Introduction

In this paper we shall consider the complementary Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M4">View MathML</a> and F is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term F involves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2">View MathML</a>, a derivative of the dependent variable. Most research papers on boundary value problems consider nonlinear terms that involve y only, and derivative-dependent nonlinearities are seldom tackled as special techniques are required.

The complementary Lidstone interpolation and boundary value problems have been very recently introduced in [1], and drawn on by Agarwal et al. in [2,3] where they consider an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M6">View MathML</a>th order differential equation together with boundary data at the odd order derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M7">View MathML</a>

(1.2)

The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4-7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an 2mth order differential equation and the Lidstone boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M8">View MathML</a>

(1.3)

There is a vast literature on Lidstone interpolation and boundary value problems. In fact, the Lidstone interpolation was first introduced by Lidstone [8] in 1929 and further characterized in the work of [9-16]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in [1,17-23]. Meanwhile, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,24-37] and the references cited therein. In most of these works the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of papers [30,31,34,35] tackle nonlinear terms that involve even order derivatives. In the present work, our study of the complementary Lidstone boundary value problem (1.1) where F depends on a derivative certainly extends and complements the rich literature on boundary value problems and notably on Lidstone boundary value problems. The literature on complementary Lidstone boundary value problems pales in comparison with that on Lidstone boundary value problems - after the first work [2] on complementary Lidstone boundary value problems, the recent paper [38] discusses the eigenvalue problem, while in [39] the existence of at least one or two positive solutions of the complementary Lidstone boundary value problem is derived by Leray-Schauder alternative and Krasnosel’skii’s fixed point theorem in a cone.

In the present work, we shall establish the existence of at least three positive solutions using fixed point theorems of Leggett and Williams [40] as well as of Avery [41]. Estimates on the norms of these solutions will also be provided. Besides achieving new results, we also compare the results in terms of generality and illustrate the importance of the results through some examples. As remarked earlier, the presence of the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M2">View MathML</a> in the nonlinear term F requires a special technique to tackle the problem.

The paper is organized as follows. Section 2 contains the necessary definitions and fixed point theorems. The existence criteria are developed and discussed in Section 3. Finally, examples are presented in Section 4 to illustrate the importance of the results obtained.

2 Preliminaries

In this section we shall state some necessary definitions, the relevant fixed point theorems and properties of certain Green’s function. Let B be a Banach space equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M10">View MathML</a>.

Definition 2.1 Let C (⊂B) be a nonempty closed convex set. We say that C is a cone provided the following conditions are satisfied:

(a) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M12">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M13">View MathML</a>;

(b) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M15">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M16">View MathML</a>.

Definition 2.2 Let C (⊂B) be a cone. A map ψ is a nonnegative continuous concave functional on C if the following conditions are satisfied:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M17">View MathML</a> is continuous;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M20">View MathML</a>.

Definition 2.3 Let C (⊂B) be a cone. A map β is a nonnegative continuous convex functional on C if the following conditions are satisfied:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M21">View MathML</a> is continuous;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M22">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M20">View MathML</a>.

Let γ, β, Θ be nonnegative continuous convex functionals on C and α, ψ be nonnegative continuous concave functionals on C. For nonnegative numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M26">View MathML</a>, we shall introduce the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M27">View MathML</a>

The following fixed point theorems are our main tools, the first is usually called Leggett-Williamsfixed point theorem, and the second is known as the five-functional fixed point theorem.

Theorem 2.1[40]

LetC (⊂B) be a cone, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M28">View MathML</a>be given. Assume thatψis a nonnegative continuous concave functional onCsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M29">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M30">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M31">View MathML</a>be a continuous and completely continuous operator. Suppose that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M34">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M35">View MathML</a>, such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M36">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M39">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M40">View MathML</a>;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43">View MathML</a>.

ThenShas (at least) three fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M46">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M47">View MathML</a>. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M48">View MathML</a>

(2.1)

Theorem 2.2[41]

LetC (⊂B) be a cone. Assume that there exist positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M49">View MathML</a>, M, nonnegative continuous convex functionalsγ, β, Θ onC, and nonnegative continuous concave functionalsα, ψonC, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M50">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M51">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M52">View MathML</a>be a continuous and completely continuous operator. Suppose that there exist nonnegative numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M54">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M55">View MathML</a>such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M56">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M58">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M59">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M61">View MathML</a>;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M63">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64">View MathML</a>;

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67">View MathML</a>.

ThenShas (at least) three fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M46">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71">View MathML</a>. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M72">View MathML</a>

(2.2)

We also require the definition of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73">View MathML</a>-Carathéodory function.

Definition 2.4[42]

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M74">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73">View MathML</a>-Carathéodory function if the following conditions hold:

(a) The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M76">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M77">View MathML</a>.

(b) The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M78">View MathML</a> is continuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a>.

(c) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M80">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M81">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M82">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M83">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a>.

To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85">View MathML</a> be the Green’s function of the Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M86">View MathML</a>

(2.3)

The Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85">View MathML</a> can be expressed as [4,5]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M88">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M89">View MathML</a>

Further, it is known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M90">View MathML</a>

(2.5)

The following two lemmas give the upper and lower bounds of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M91">View MathML</a>, they play an important role in subsequent development. We remark that the bounds in the two lemmas are sharper than those given in the literature [4,5,35,37].

Lemma 2.1[38]

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M92">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M93">View MathML</a>

Lemma 2.2[38]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M94">View MathML</a>be given. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M95">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M96">View MathML</a>

3 Triple positive solutions

In this section, we shall use the fixed point theorems stated in Section 2 to obtain the existence of at least three positive solutions of the complementary Lidstone boundary value problem (1.1). By a positive solutiony of (1.1), we mean a nontrivial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M97">View MathML</a> satisfying (1.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M98">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99">View MathML</a>.

To tackle (1.1), we first consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M100">View MathML</a>

(3.1)

whose solution is simply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M101">View MathML</a>

(3.2)

Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M102">View MathML</a>

(3.3)

If (3.3) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103">View MathML</a>, then by virtue of (3.2), the boundary value problem (1.1) has a solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M104">View MathML</a>

(3.4)

So the existence of a solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a solution of the Lidstone boundary value problem (3.3). It is clear from (3.4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M105">View MathML</a>; moreover if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103">View MathML</a> is positive, so is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M107">View MathML</a>. With the tools in Section 2 and a technique to handle the nonlinear term F, we shall study the boundary value problem (1.1) via (3.3).

Let the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M108">View MathML</a> be equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M109">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M110">View MathML</a>. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M111">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M112">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M85">View MathML</a> is the Green’s function given in (2.4). A fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M103">View MathML</a> of the operator S is clearly a solution of the boundary value problem (3.3), and as seen earlier <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M115">View MathML</a> is a solution of (1.1).

For easy reference, we shall list the conditions that are needed later. In these conditions the sets K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M116">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M117">View MathML</a>

(3.6)

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M118">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73">View MathML</a>-Carathéodory function.

(C2) We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M120">View MathML</a>

(C3) There exist continuous functions f, ν, μ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M122">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M123">View MathML</a>

(C4) There exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M124">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M125">View MathML</a>

If (C2) and (C3) hold, then it follows from (3.5) that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M128">View MathML</a>

(3.7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a> be fixed. We define a cone C in B as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M130">View MathML</a>

(3.8)

where θ is given in (C4). Clearly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M131">View MathML</a>.

Lemma 3.1Let (C1)-(C4) hold. Then the operatorSdefined in (3.5) is continuous and completely continuous, andSmapsCintoC.

Proof From (2.4) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a> and the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M134">View MathML</a> is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M136">View MathML</a>. This together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M137">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M73">View MathML</a>-Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that S is continuous and completely continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a>. From (3.7) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M140">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a>. Next, using (3.7) and Lemma 2.1 gives for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M143">View MathML</a>

(3.9)

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M144">View MathML</a>

(3.10)

Now, employing (3.7), Lemma 2.2, (C4) and (3.10), we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M146">View MathML</a>

This leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M147">View MathML</a>

We have shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M148">View MathML</a>. □

For subsequent results, we define the following constants for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M150">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M151">View MathML</a>

(3.11)

Lemma 3.2Let (C1)-(C4) hold, and assume

(C5) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135">View MathML</a>of positive measure.

Suppose that there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M154">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M155">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M156">View MathML</a>

(3.12)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M157">View MathML</a>

(3.13)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M158">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M159">View MathML</a>, which implies immediately that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M160">View MathML</a>

Then, using (3.9), (C5) and (3.12), we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M162">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M163">View MathML</a>. Together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M148">View MathML</a> (Lemma 3.1), we have shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M165">View MathML</a>. Conclusion (3.13) is now immediate. □

Using a similar argument as Lemma 3.2, we have the following lemma.

Lemma 3.3Let (C1)-(C4) hold. Suppose that there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M154">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M167">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M168">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M169">View MathML</a>

We are now ready to establish the existence of three positive solutions for the complementary Lidstone boundary value problem (1.1). The first result below uses Leggett-Williams’ fixed point theorem (Theorem 2.1).

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a>be fixed. Let (C1)-(C5) hold, and assume

(C6) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M173">View MathML</a>of positive measure.

Suppose that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M34">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M177">View MathML</a>

such that the following hold:

(P) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M178">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M179">View MathML</a>;

(Q) one of the following holds:

(Q1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M180">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M181">View MathML</a>;

(Q2) there exists a numberd (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M182">View MathML</a>) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M183">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M167">View MathML</a>;

(R) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M185">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M186">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M187">View MathML</a>.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M188">View MathML</a> (whereCis defined in (3.8)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M189">View MathML</a>

(3.14)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M194">View MathML</a>

(3.15)

(where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M195">View MathML</a>s are those in conclusion (a)). We further have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M196">View MathML</a>

(3.16)

Proof We shall employ Theorem 2.1 with the cone C defined in (3.8). First, we shall prove that condition (Q) implies the existence of a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M197">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M198">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M199">View MathML</a>

(3.17)

Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M200">View MathML</a>. Suppose now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M201">View MathML</a> of (Q1) is satisfied. Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M203">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M204">View MathML</a>

(3.18)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M205">View MathML</a>

Noting (3.18), it is then clear that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M206">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M207">View MathML</a>

(3.19)

Now, pick the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M197">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M209">View MathML</a>

(3.20)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M210">View MathML</a>. Using (3.9), (3.19) and (3.20) yields for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M99">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M212">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M213">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M214">View MathML</a>. Thus, (3.17) follows immediately. Note that the argument is similar if we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M215">View MathML</a> of (Q1) is satisfied.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M17">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M217">View MathML</a>

Clearly, ψ is a nonnegative continuous concave functional on C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M29">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a>.

We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M220">View MathML</a>

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M36">View MathML</a>. Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M224">View MathML</a> which imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M225">View MathML</a>

(3.21)

Using (3.7), (3.21), (C6) and (R), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M226">View MathML</a>

Therefore, we have shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M38">View MathML</a>.

Next, by condition (P) and Lemma 3.2 (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M229">View MathML</a>), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M230">View MathML</a>. Hence, condition (b) of Theorem 2.1 is fulfilled.

Finally, we shall show that condition (c) of Theorem 2.1 holds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43">View MathML</a>. Using (3.7), Lemma 2.2, (C4), (3.10) and the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M233">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M234">View MathML</a>

Hence, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M37">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M42">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M43">View MathML</a>.

It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M238">View MathML</a> satisfying (2.1). It is easy to see that here (2.1) reduces to (3.14). This completes the proof of conclusion (a).

Finally, it is observed from (3.4) that the complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M243">View MathML</a>

(3.22)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M244">View MathML</a>, we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M246">View MathML</a>

(3.23)

Combining (3.22) and (3.23) gives (3.15) immediately.

Further, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M247">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M250">View MathML</a>

(3.24)

Hence, noting (3.14), (3.15) and (3.24), we get (3.16). This completes the proof of conclusion (b). □

We shall now employ the five-functional fixed point theorem (Theorem 2.2) to give other existence criteria. In applying Theorem 2.2 it is possible to choose the functionals and constants in different ways, indeed we shall do so and derive two results. Our first result below turns out to be a generalization of Theorem 3.1.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a>be fixed. Let (C1)-(C4) hold. Assume that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M253">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M254">View MathML</a>

such that

(C7) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M257">View MathML</a>of positive measure;

(C8) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M259">View MathML</a>of positive measure.

Suppose that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M261">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M262">View MathML</a>

such that the following hold:

(P) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M263">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M264">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M265">View MathML</a>;

(Q) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267">View MathML</a>;

(R) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M268">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M269">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270">View MathML</a>.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M271">View MathML</a> (whereCis defined in (3.8)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M272">View MathML</a>

(3.25)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a>such that (3.15) holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193">View MathML</a>. We further have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M277">View MathML</a>

(3.26)

Proof We shall apply Theorem 2.2 with the cone C defined in (3.8). We define the following five functionals on the cone C:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M278">View MathML</a>

(3.27)

First, we shall show that the operator S maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M71">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M281">View MathML</a>. By (Q) and Lemma 3.3 (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M282">View MathML</a>), we immediately have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M283">View MathML</a>.

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M284">View MathML</a>

since it has an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M285">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M286">View MathML</a>. Then by definition we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M287">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M288">View MathML</a>, which imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M289">View MathML</a>

(3.28)

Noting (3.7), (3.28), (C7) and (R), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M290">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M286">View MathML</a>.

We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M294">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M295">View MathML</a>

because it has an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M296">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M300">View MathML</a>

(3.29)

which lead to the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M301">View MathML</a>

(3.30)

Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M302">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297">View MathML</a>.

Next, we shall show that condition (c) of Theorem 2.2 is met. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a>. Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M306">View MathML</a>

(3.31)

Moreover, using the fact that S maps C into C, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M307">View MathML</a>

(3.32)

Combining (3.31) and (3.32) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M308">View MathML</a>

(3.33)

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M309">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64">View MathML</a>. Then it follows from (3.33) and the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M311">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M312">View MathML</a>

(3.34)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M57">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M309">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M64">View MathML</a>.

Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M316">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299">View MathML</a> which give (3.29) and (3.30). As in proving condition (b), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a>. Hence, condition (d) of Theorem 2.2 is satisfied.

It now follows from Theorem 2.2 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M321">View MathML</a> satisfying (2.2). Furthermore, (2.2) reduces to (3.25) immediately. This completes the proof of conclusion (a).

Finally, as in the proof of Theorem 3.1, we see that (3.15) holds for the positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193">View MathML</a>, of the complementary Lidstone boundary value problem (1.1). Moreover, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M324">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M325">View MathML</a>, we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M325">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M327">View MathML</a>

Next, noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M328">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255">View MathML</a>, we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M255">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M331">View MathML</a>

Lastly, using (3.15) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M332">View MathML</a>, we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M333">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M334">View MathML</a>

The proof of conclusion (b) is complete. □

We shall now consider the special case of Theorem 3.2 when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M335">View MathML</a>

Then, from definitions (3.11), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M336">View MathML</a>

In this case Theorem 3.2 yields the following corollary.

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a>be fixed. Let (C1)-(C4) hold, and assume

(C7)′ for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M145">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M172">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M173">View MathML</a>of positive measure;

(C8)′ the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M152">View MathML</a>on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M135">View MathML</a>of positive measure.

Suppose that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M261">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M345">View MathML</a>

such that the following hold:

(P) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M346">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M347">View MathML</a>;

(Q) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267">View MathML</a>;

(R) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M350">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M351">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270">View MathML</a>.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M271">View MathML</a> (whereCis defined in (3.8)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M354">View MathML</a>

(3.35)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a>such that (3.15) holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M193">View MathML</a>. We further have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M359">View MathML</a>

(3.36)

Remark 3.1 Corollary 3.1 is actually Theorem 3.1. Since Corollary 3.1 is a special case of Theorem 3.2, this shows that Theorem 3.2 is more general than Theorem 3.1.

The next theorem illustrates another application of Theorem 2.2. Compared to the conditions in Theorem 3.2, here the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M362">View MathML</a> have different ranges and condition (P) is also different. Note that in the proof of Theorem 3.3 the functionals ψ and Θ are chosen differently from those in Theorem 3.2.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M129">View MathML</a>be fixed. Let (C1)-(C4) hold. Assume that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M253">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M366">View MathML</a>

such that (C7) and (C8) hold. Suppose that there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M368">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M369">View MathML</a>

such that the following hold:

(P) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M263">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M264">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M372">View MathML</a>;

(Q) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M266">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M267">View MathML</a>;

(R) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M268">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M269">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M270">View MathML</a>.

Then we have conclusions (a) and (b) of Theorem 3.2.

Proof To apply Theorem 2.2, we shall define the following functionals on the cone C (see (3.8)):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M378">View MathML</a>

(3.37)

As in the proof of Theorem 3.2, using (Q) and Lemma 3.3 we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M379">View MathML</a>.

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we use (R) and a similar argument as in the proof of Theorem 3.2.

We shall now prove that condition (b) of Theorem 2.2 is satisfied. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M380">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M297">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M299">View MathML</a> which imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M385">View MathML</a>

(3.38)

and also (3.30). In view of (3.9), (3.38), (3.30), (C8), (P) and (Q), we obtain, as in the proof of Theorem 3.2, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a>. Therefore, condition (b) of Theorem 2.2 is fulfilled.

Next, using a similar argument as in the proof of Theorem 3.2, we see that condition (c) of Theorem 2.2 is met.

Finally, we shall verify that condition (d) of Theorem 2.2 is fulfilled. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M11">View MathML</a>. It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M388">View MathML</a>

(3.39)

Noting that S maps C into C, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M389">View MathML</a>

(3.40)

A combination of (3.39) and (3.40) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M390">View MathML</a>

(3.41)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67">View MathML</a>. Then (3.41) and the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M393">View MathML</a> lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M394">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M60">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M66">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M67">View MathML</a>.

Conclusion (a) now follows from Theorem 2.2 immediately, while conclusion (b) is similarly obtained as in Theorem 3.2. □

4 Examples

In this section, we shall present examples to illustrate the usefulness as well as to compare the generality of the results obtained in Section 3.

Example 4.1 Consider the complementary Lidstone boundary value problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398">View MathML</a> and the nonlinear term F given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M399">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M400">View MathML</a> is continuous in each argument and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M401">View MathML</a>

(4.2)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M402">View MathML</a> is fixed and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s and d are in the context of Theorem 3.1 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M404">View MathML</a>

(4.3)

Let the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M405">View MathML</a> (which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M406">View MathML</a>). Then it is clear that (C1)-(C6) are fulfilled. Moreover, by direct computation we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M407">View MathML</a>, and on using Lemma 2.2 we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M408">View MathML</a>

For convenience, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M409">View MathML</a> although this will lead to more stringent conditions.

Hence, (4.3) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M410">View MathML</a>

(4.4)

and clearly we can easily find numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s and d that satisfy (4.4).

We shall check the conditions of Theorem 3.1. First, condition (P) is obviously satisfied. Next, from (4.3) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M412">View MathML</a>, therefore for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M413">View MathML</a> it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M414">View MathML</a>

Hence, condition (Q2) is met. Finally, (R) is satisfied since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M415">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M416">View MathML</a>

By Theorem 3.1 (conclusion (b)), the boundary value problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M402">View MathML</a>, (4.1) and (4.2) has (at least) three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a> such that (from (3.16))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M422">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s satisfy (4.4).

Example 4.2 Consider the complementary Lidstone boundary value problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398">View MathML</a> and the nonlinear term F given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M425">View MathML</a>

(4.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M426">View MathML</a> is continuous in each argument and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M427">View MathML</a>

(4.7)

Here, we fix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M428">View MathML</a>

(4.8)

and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s are in the context of Theorem 3.2 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M430">View MathML</a>

(4.9)

Let the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M405">View MathML</a> (which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M406">View MathML</a>). Then it is clear that (C1)-(C4), (C7) and (C8) are fulfilled. Moreover, by direct computation we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M433">View MathML</a>

By using Lemma 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M434">View MathML</a>

For convenience, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M435">View MathML</a> although this will lead to more stringent conditions. Hence, (4.9) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M436">View MathML</a>

or equivalently (combining the first two inequalities)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M437">View MathML</a>

(4.10)

It is clear that we can easily find numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s that fulfill (4.10).

We shall check the conditions of Theorem 3.2. First, condition (P) is obviously satisfied. Next, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M439">View MathML</a>

(4.11)

we find for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M440">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M441">View MathML</a>

Hence, condition (Q) is met. Finally, (R) is satisfied since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M442">View MathML</a>, using (4.11) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M443">View MathML</a>

By Theorem 3.2 (conclusion (b)), the boundary value problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M398">View MathML</a>, (4.6), (4.7) and (4.8) has (at least) three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M192">View MathML</a> such that (from (3.26))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M448">View MathML</a>

(4.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M25">View MathML</a>’s satisfy (4.10).

Remark 4.1 In Example 4.2, we see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M442">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/125/mathml/M451">View MathML</a>

Thus, condition (R) of Corollary 3.1 is not satisfied and so Corollary 3.1 cannot be used to establish the existence of triple positive solutions in Example 4.2. Recalling that Corollary 3.1 is actually Theorem 3.1, this illustrates the case when Theorem 3.2 is applicable but not Theorem 3.1. Hence, this example shows that Theorem 3.2 is indeed more general than Theorem 3.1.

Competing interests

The author declares that she has no competing interests.

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