Open Access Research

Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions

Tadeusz Jankowski

Author Affiliations

Department of Differential Equations and Applied Mathematics, Gdańsk University of Technology, 11/12 G. Narutowicz Str., Gdańsk, 80-233, Poland

Boundary Value Problems 2013, 2013:8  doi:10.1186/1687-2770-2013-8


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/8


Received:20 September 2012
Accepted:15 January 2013
Published:17 January 2013

© 2013 Jankowski; 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

In this paper, we consider the existence of positive solutions for second-order differential equations with deviating arguments and nonlocal boundary conditions. By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions under which such boundary value problems have at least three positive solutions. We discuss our problem both for delayed and advanced arguments α and also in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2">View MathML</a>. In all cases, the argument β can change the character on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M3">View MathML</a>, see problem (1). It means that β can be delayed in some set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M4">View MathML</a> and advanced in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M5">View MathML</a>. An example is added to illustrate the results.

MSC: 34B10.

Keywords:
boundary value problems with delayed and advanced arguments; nonlocal boundary conditions; cone; existence of positive solutions; a fixed point theorem

1 Introduction

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M7">View MathML</a>. Let us consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M8">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> denote linear functionals on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M11">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M12">View MathML</a>

involving Stieltjes integrals with suitable functions A and B of bounded variation on J. It is not assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> are positive to all positive x. As we see later, the measures dA, dB can be signed measures.

We introduce the following assumptions:

H1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M16">View MathML</a>, A and B are functions of bounded variation;

H2: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M17">View MathML</a> and h does not vanish identically on any subinterval;

H3: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M18">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M19">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M22">View MathML</a>.

Recently, the existence of multiple positive solutions for differential equations has been studied extensively; for details, see, for example, [1-31]. However, many works about positive solutions have been done under the assumption that the first-order derivative is not involved explicitly in nonlinear terms; see, for example, [3,6,8-14,17,20,25-27,30]. From this list, only papers [9-12,14,20,30] concern positive solutions to problems with deviating arguments. On the other hand, there are some papers considering the multiplicity of positive solutions with dependence on the first-order derivative; see, for example, [2,4,5,7,15,16,18,19,21-24,28,29,31]. Note that boundary conditions (BCs) in differential problems have important influence on the existence of the results obtained. In this paper, we consider problem (1) which is a problem with dependence on the first-order derivative with BCs involving Stieltjes integrals with signed measures of dA, dB appearing in functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a>; moreover, problem (1) depends on deviating arguments.

For example, in papers [2,4,15,18,22,24], the existence of positive solutions to second-order differential equations with dependence on the first-order derivative (but without deviating arguments) has been studied with various BCs including the following:

by fixed point theorems in a cone (such as Avery-Peterson, an extension of Krasnoselskii’s fixed point theorem or monotone iterative method) with corresponding assumptions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M26">View MathML</a>

or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M27">View MathML</a>, respectively.

For example, in papers [8-11,20,22,30], the existence of positive solutions to second-order differential equations including impulsive problems, but without dependence on the first-order derivative, has been studied with various BCs including the following:

under corresponding assumptions by fixed point theorems in a cone (such as Avery-Peterson, Leggett-Williams, Krasnoselskii or fixed point index theorem). See also paper [13], where positive solutions have been discussed for second-order impulsive problems with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M29">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a> has the same form as in problem (1) with signed measure dA appearing in functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>.

Positive solutions to second-order differential equations with boundary conditions that involve Stieltjes integrals have been studied in the case of signed measures in papers [25,26] with BCs including, for example, the following:

The main results of papers [25,26] have been obtained by the fixed point index theory for problems without deviating arguments. The study of positive solutions to boundary value problems with Stieltjes integrals in the case of signed measures has also been done in papers [3,7,13,14,27] for second-order differential equations (also impulsive) or third-order differential equations by using the fixed point index theory, the Avery-Peterson fixed point theorem or fixed point index theory involving eigenvalues.

Note that BCs in problem (1) with functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> cover some nonlocal BCs, for example,

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M37">View MathML</a> and some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39">View MathML</a>. In our paper, the assumption that the measures dA, dB in the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> are positive is not needed. More precisely, one needs to choose the above functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39">View MathML</a> in such a way that the assumption H4 holds. It means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M39">View MathML</a> can change sign on J.

A standard approach (see, for example, [25-27]) to studying positive solutions of boundary value problems such as (1) is to translate problem (1) to a Hammerstein integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M46">View MathML</a>

(2)

to find a solution as a fixed point of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M47">View MathML</a> by using a fixed point theorem in a cone. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M50">View MathML</a> are corresponding continuous functions while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> have the same form as in problem (1). G denotes a Green function connected with our problem, so in our case it is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M53">View MathML</a>

In our paper, we eliminate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> from problem (2) to obtain the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M56">View MathML</a> with a corresponding operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M57">View MathML</a>, and then we seek solutions as fixed points of this operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M57">View MathML</a>.

Note that if we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a> in the BCs of problem (1), then this new problem is more general than the previous one because in this case someone, for example, can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M61">View MathML</a>. In this paper, we try to explain why for some cases we have to discuss problem (1) with constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M62">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63">View MathML</a>.

To apply such a fixed point theorem in a cone to problem (1), we have to construct a suitable cone K. Usually, we need to find a nonnegative function κ and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M65">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M67">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69">View MathML</a> (see, for example, [25-27]) to work with the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M70">View MathML</a>

Indeed, for problems without deviating arguments, someone can use any interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M71">View MathML</a>. It means that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a> on J, then we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a> in the boundary conditions of problem (1) to work with the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M74">View MathML</a>

for ζ, ϱ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M76">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M77">View MathML</a>; see Section 5.

Note that for problems with delayed or advanced arguments, we have to use interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M78">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M79">View MathML</a>, respectively. We see that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M81">View MathML</a> for problem (1) with deviated arguments. It shows that the approach from papers [25-27] needs a little modification to problems with delayed or advanced arguments. Consider the situation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M82">View MathML</a> on J. In this case, we can put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83">View MathML</a> in the boundary conditions of problem (1) to find a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M84">View MathML</a> to work with the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M85">View MathML</a>

see Section 3. For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86">View MathML</a> on J, we can put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87">View MathML</a> to work similarly as in Section 3; see Section 4. Note that in the above three cases for the argument β, we need only the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M88">View MathML</a>, which means that β can change the character in J.

Note that in cited papers, positive solutions to differential equations with dependence on the first-order derivative have been investigated only for problems without deviating arguments, see [2,4,5,7,15,16,18,19,21-24,28,29,31]. Moreover, BCs in problem (1) cover some nonlocal BCs discussed earlier.

Motivated by [25-27], in this paper, we apply the fixed point theorem due to Avery-Peterson to obtain sufficient conditions for the existence of multiple positive solutions to problems of type (1). In problem (1), an unknown x depends on deviating arguments which can be both of advanced or delayed type. To the author’s knowledge, it is the first paper when positive solutions have been investigated for such general boundary value problems with functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a> and with deviating arguments α, β in differential equations in which f depends also on the first-order derivative. It is important to indicate that problems of type (1) have been discussed with signed measures of dA, dB appearing in Stieltjes integrals of functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M10">View MathML</a>.

The organization of this paper is as follows. In Section 2, we present some necessary lemmas connected with our main results. In Section 3, we first present some definitions and a theorem of Avery and Peterson which is useful in our research. Also in Section 3, we discuss the existence of multiple positive solutions to problems with delayed argument α, by using the above mentioned Avery-Peterson theorem. At the end of this section, an example is added to verify theoretical results. In Section 4, we formulate sufficient conditions under which problems with advanced argument α have positive solutions. In the last section, we discuss problems of type (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a> on J.

2 Some lemmas

Let us introduce the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M94">View MathML</a>

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. Assume thatAandBare functions of bounded variation and, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M98">View MathML</a>

with

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M99">View MathML</a>or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M100">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M101">View MathML</a>

Here, VarAdenotes the variation of a functionAonJ.

Proof Note that in case (i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M102">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M103">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M104">View MathML</a>

Combining this with the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M105">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M106">View MathML</a>

This proves case (i).

In case (ii), similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M107">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M108">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M109">View MathML</a>

Adding to this the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M110">View MathML</a>

we get the result in case (ii). This ends the proof. □

Remark 1 If we assume that A and B are increasing functions, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M111">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M112">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M113">View MathML</a>

Similarly, we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M114">View MathML</a>

Now, the constant M from Lemma 1 has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M115">View MathML</a>

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M116">View MathML</a>

(3)

Let us introduce the assumption.

H0: A and B are functions of bounded variation and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M117">View MathML</a>

for

We require the following result.

Lemma 2Let the assumption H0hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M119">View MathML</a>. Then problem (3) has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M120">View MathML</a>

with

Proof Integrating the differential equation in (3) two times, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M122">View MathML</a>

(4)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M123">View MathML</a> and use the boundary conditions from problem (3) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M124">View MathML</a>

Now, finding from this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M125">View MathML</a> and then substituting it to formula (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M126">View MathML</a>

(5)

Next, putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M127">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M128">View MathML</a>, and then substitute it to formula (5) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M129">View MathML</a>

(6)

Now, we have to eliminate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M131">View MathML</a> from (6). If u is a solution of (6), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M132">View MathML</a>

Solving this system with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M131">View MathML</a> and then substituting to (6), we have the assertion of this lemma. This ends the proof. □

Define the operator T by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M135">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M136">View MathML</a>

We consider the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M137">View MathML</a> with the maximum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M138">View MathML</a>. Define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M139">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M140">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M141">View MathML</a>

Let us introduce the following assumption.

H4: A and B are functions of bounded variation and

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M146">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M147">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M150">View MathML</a>, δ, Δ are defined as in the assumption H0,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M160">View MathML</a>.

Lemma 3Let the assumptions H1-H4hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M161">View MathML</a>.

Proof Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M162">View MathML</a> is a positive solution of problem (1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M162">View MathML</a> solves the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M164">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M165">View MathML</a>

(7)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M167">View MathML</a> in view of the assumptions H1, H2, H4 and the positivity of Green’s function G.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M168">View MathML</a>. Moreover,

Hence, Tu is concave and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M170">View MathML</a> on J.

We next show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M172">View MathML</a>. Indeed,

Finally, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M174">View MathML</a>

To do it, we consider two steps. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M175">View MathML</a>.

Step 1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M176">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M179">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M181">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M182">View MathML</a>

It yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M183">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M184">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M185">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M186">View MathML</a>

It yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M187">View MathML</a>

Step 2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M188">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M190">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M191">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M192">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M193">View MathML</a>

It shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M161">View MathML</a>. This ends the proof. □

Remark 2 Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M196">View MathML</a>. Note that the measure changes the sign and is increasing. It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M197">View MathML</a>

If we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M198">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69">View MathML</a>.

Remark 3 Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M204">View MathML</a>. Note that the measure changes the sign and is increasing. It is easy to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M205">View MathML</a>

If we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M206">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69">View MathML</a>.

Remark 4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. Then the assumptions H3, H4 hold if one of the following conditions is satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M215">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M217">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a>.

We consider only case (i). First of all, we see that dA, dB change the sign and are increasing. Indeed, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M221">View MathML</a>

It means that the assumption H3 holds. Moreover,

It proves that the assumption H4 holds.

By a similar way, we prove the assertion in case (ii) or (iii).

3 Positive solutions to problem (1) with delayed arguments

Now, we present the necessary definitions from the theory of cones in Banach spaces.

Definition 1 Let E be a real Banach space. A nonempty convex closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M223">View MathML</a> is said to be a cone provided that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M224">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M225">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M226">View MathML</a>, and

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M227">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M228">View MathML</a>.

Note that every cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M223">View MathML</a> induces an ordering in E given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M230">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M231">View MathML</a>.

Definition 2 A map Φ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M232">View MathML</a> is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M233">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2">View MathML</a>.

Similarly, we say the map φ is a nonnegative continuous convex functional on a cone P of a real Banach space E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M236">View MathML</a> is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M237">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M2">View MathML</a>.

Definition 3 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Let φ and Θ be nonnegative continuous convex functionals on P, let Φ be a nonnegative continuous concave functional on P, and let Ψ be a nonnegative continuous functional on P. Then, for positive numbers a, b, c, d, we define the following sets:

We will use the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1).

Theorem 1 (see [1])

LetPbe a cone in a real Banach spaceE. Letφand Θ be nonnegative continuous convex functionals onP, let Φ be a nonnegative continuous concave functional onP, and let Ψ be a nonnegative continuous functional onPsatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M241">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M242">View MathML</a>such that for some positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M243">View MathML</a>andd,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M244">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M245">View MathML</a>. Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M246">View MathML</a>

is completely continuous and there exist positive numbersa, b, cwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247">View MathML</a>such that

(S1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M248">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M249">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M250">View MathML</a>,

(S2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M249">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M252">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M253">View MathML</a>,

(S3): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M254">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M255">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M256">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M257">View MathML</a>.

ThenThas at least three fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M258">View MathML</a>such that

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M260">View MathML</a>

We apply Theorem 1 with the cone K instead of P and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M261">View MathML</a>. Now, we define the nonnegative continuous concave functional Φ on K by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M262">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M263">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M264">View MathML</a>.

Now, we can formulate the main result of this section.

Theorem 2Let the assumptions H1-H4hold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M62">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. In addition, we assume that there exist positive constantsa, b, c, d, M, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247">View MathML</a>and such that

with

and

(A1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273">View MathML</a>,

(A2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M275">View MathML</a>,

(A3): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277">View MathML</a>.

Then problem (1) has at least three nonnegative solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M283">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284">View MathML</a>.

Proof Basing on the definitions of T, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M285">View MathML</a> is equicontinuous on J, so T is completely continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M245">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M287">View MathML</a>. By Lemma 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M288">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. Assumption (A1) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M291">View MathML</a>.

Moreover, in view of (7),

Combining it, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M293">View MathML</a>

This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M294">View MathML</a>.

Now, we need to show that condition (S1) is satisfied. Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M295">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>, and

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. Moreover,

This proves that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M302">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M303">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M305">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M307">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M304">View MathML</a>. Assumption (A2) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M309">View MathML</a>. Hence,

Moreover,

It yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M312">View MathML</a>

This proves that condition (S1) holds.

Now, we need to prove that condition (S2) is satisfied. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M314">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M315">View MathML</a>

so condition (S2) holds.

Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M316">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M254">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M256">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M319">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M322">View MathML</a>

and finally,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M323">View MathML</a>

This shows that condition (S3) is satisfied.

Since all the conditions of Theorem 1 are satisfied, problem (1) has at least three nonnegative solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M327">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M329">View MathML</a>

This ends the proof. □

Example Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M330">View MathML</a>

(8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M331">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M332">View MathML</a>. For example, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M334">View MathML</a> on J with fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M335">View MathML</a>. Indeed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M336">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M337">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M83">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M341">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M342">View MathML</a>, so the measure changes the sign on J. Moreover,

so the assumption H4 holds; see Remark 4. Next,

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M345">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M346">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M347">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M349">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M350">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M352">View MathML</a>. Then

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M354">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M355">View MathML</a>.

All the assumptions of Theorem 2 hold, so problem (8) has at least three positive solutions.

Remark 5 We can also construct an example in which, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M356">View MathML</a> to use the results of Remark 4. Note that also this measure changes the sign.

4 Positive solutions to problem (1) with advanced arguments

In this section, we consider the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86">View MathML</a> on J, so the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M358">View MathML</a> is now replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M359">View MathML</a>. It means that we can put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63">View MathML</a> in the boundary conditions of problem (1) because someone can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M362">View MathML</a> as an example. Let us introduce the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M363">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M364">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M365">View MathML</a>

Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M366">View MathML</a>. Functionals Ψ, Θ, φ are defined as in Section 3. We formulate only the main result using the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M363">View MathML</a> instead of K (see Theorem 2); the proof is similar to the previous one.

Theorem 3Let the assumptions H1-H4hold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M63">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. In addition, we assume that there exist positive constantsa, b, c, d, M, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247">View MathML</a>and such that

with

and

(B1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273">View MathML</a>,

(B2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M378">View MathML</a>,

(B3): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277">View MathML</a>.

Then problem (1) has at least three nonnegative solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M386">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284">View MathML</a>.

5 Positive solutions to problem (1) for the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a> on J

In this section, we consider problem (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a> on J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a>. It means that now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M391">View MathML</a> for some fixed constants ζ, ϱ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M76">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M393">View MathML</a> we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M394">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M395">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M69">View MathML</a>. Now, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M77">View MathML</a>, we introduce the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M398">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M399">View MathML</a>

Functionals Ψ, Θ, φ are defined as in Section 3; the cone K is now replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M398">View MathML</a>.

Theorem 4Let the assumptions H1-H4hold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M59">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M393">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M21">View MathML</a>. In addition, we assume that there exist positive constantsa, b, c, d, M, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M247">View MathML</a>and such that

with

and

(C1): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M272">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M273">View MathML</a>,

(C2): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M274">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M411">View MathML</a>,

(C3): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M276">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M277">View MathML</a>.

Then problem (1) has at least three nonnegative solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M280">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M282">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M419">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/8/mathml/M284">View MathML</a>.

6 Conclusions

In this paper, we have discussed boundary value problems for second-order differential equations with deviating arguments and with dependence on the first-order derivative. In our research, the deviating arguments can be both delayed and advanced. By using the fixed point theorem of Avery and Peterson, new sufficient conditions for the existence of positive solutions to such boundary problems have been derived. An example is provided for illustration.

Competing interests

The author declares that he has no competing interests.

References

  1. Avery, RI, Peterson, AC: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl.. 42, 313–322 (2001). Publisher Full Text OpenURL

  2. Bai, Z, Ge, W: Existence of three positive solutions for some second-order boundary value problems. Comput. Math. Appl.. 48, 699–707 (2004). Publisher Full Text OpenURL

  3. Graef, JR, Webb, JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Anal.. 71, 1542–1551 (2009). Publisher Full Text OpenURL

  4. Guo, Y, Ge, W: Positive solutions for three-point boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl.. 290, 291–301 (2004). Publisher Full Text OpenURL

  5. Guo, Y, Yu, C, Wang, J: Existence of three positive solutions for m-point boundary value problems on infinite intervals. Nonlinear Anal.. 71, 717–722 (2009). Publisher Full Text OpenURL

  6. Infante, G, Pietramala, P: Nonlocal impulsive boundary value problems with solutions that change sign. In: Cabada A, Liz E, Nieto JJ (eds.) Proceedings of the International Conference on Boundary Value Problems. (2009)

  7. Infante, G, Pietramala, P, Zima, M: Positive solutions for a class of nonlocal impulsive BVPs via fixed point index. Topol. Methods Nonlinear Anal.. 36, 263–284 (2010)

  8. Jankowski, T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput.. 202, 550–561 (2008). Publisher Full Text OpenURL

  9. Jankowski, T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Appl. Math. Comput.. 197, 179–189 (2008). Publisher Full Text OpenURL

  10. Jankowski, T: Existence of positive solutions to second order four-point impulsive differential equations with deviating arguments. Comput. Math. Appl.. 58, 805–817 (2009). Publisher Full Text OpenURL

  11. Jankowski, T: Three positive solutions to second-order three-point impulsive differential equations with deviating arguments. Int. J. Comput. Math.. 87, 215–225 (2010). Publisher Full Text OpenURL

  12. Jankowski, T: Multiple solutions for a class of boundary value problems with deviating arguments and integral boundary conditions. Dyn. Syst. Appl.. 19, 179–188 (2010)

  13. Jankowski, T: Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. Nonlinear Anal.. 74, 3775–3785 (2011). Publisher Full Text OpenURL

  14. Jankowski, T: Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions. Nonlinear Anal.. 75, 913–923 (2012). Publisher Full Text OpenURL

  15. Ji, D, Ge, W: Multiple positive solutions for some p-Laplacian boundary value problems. Appl. Math. Comput.. 187, 1315–1325 (2007). Publisher Full Text OpenURL

  16. Ji, D, Ge, W: Existence of multiple positive solutions for one-dimensional p-Laplacian operator. J. Appl. Math. Comput.. 26, 451–463 (2008). Publisher Full Text OpenURL

  17. Karakostos, GL, Tsamatos, PC: Existence of multipoint positive solutions for a nonlocal boundary value problem. Topol. Methods Nonlinear Anal.. 19, 109–121 (2002)

  18. Sun, B, Qu, Y, Ge, W: Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem. Appl. Math. Comput.. 197, 389–398 (2008). Publisher Full Text OpenURL

  19. Sun, B, Ge, W, Zhao, D: Three positive solutions for multipoint one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative. Math. Comput. Model.. 45, 1170–1178 (2007). Publisher Full Text OpenURL

  20. Wang, W, Sheng, J: Positive solutions to a multi-point boundary value problem with delay. Appl. Math. Comput.. 188, 96–102 (2007). Publisher Full Text OpenURL

  21. Wang, Y, Ge, W: Multiple positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian. J. Math. Anal. Appl.. 327, 1381–1395 (2007). Publisher Full Text OpenURL

  22. Wang, Y, Ge, W: Existence of triple positive solutions for multi-point boundary value problems with one-dimensional p-Laplacian. Comput. Math. Appl.. 54, 793–807 (2007). Publisher Full Text OpenURL

  23. Wang, Y, Zhao, W, Ge, W: Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian. J. Math. Anal. Appl.. 326, 641–654 (2007). Publisher Full Text OpenURL

  24. Wang, Z, Zhang, J: Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl.. 314, 618–630 (2006). Publisher Full Text OpenURL

  25. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc.. 74, 673–693 (2006). Publisher Full Text OpenURL

  26. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl.. 15, 45–67 (2008). Publisher Full Text OpenURL

  27. Webb, JRL, Infante, G: Non-local boundary value problems of arbitrary order. J. Lond. Math. Soc.. 79, 238–259 (2009)

  28. Yan, B, O’Regan, D, Agarwal, RP: Multiple positive solutions of singular second order boundary value problems with derivative dependence. Aequ. Math.. 74, 62–89 (2007). Publisher Full Text OpenURL

  29. Yang, L, Liu, X, Jia, M: Multiplicity results for second-order m-point boundary value problem. J. Math. Anal. Appl.. 324, 532–542 (2006). Publisher Full Text OpenURL

  30. Yang, C, Zhai, C, Yan, J: Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal.. 65, 2013–2023 (2006). Publisher Full Text OpenURL

  31. Yang, Y, Xiao, D: On existence of multiple positive solutions for ϕ-Laplacian multipoint boundary value. Nonlinear Anal.. 71, 4158–4166 (2009). Publisher Full Text OpenURL