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Open Access Research

Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian

Xuemei Zhang1* and Meiqiang Feng2

Author Affiliations

1 Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, Republic of China

2 School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, Republic of China

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Boundary Value Problems 2014, 2014:112  doi:10.1186/1687-2770-2014-112

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/112


Received:29 January 2014
Accepted:29 April 2014
Published:13 May 2014

© 2014 Zhang and Feng; 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 credited.

Abstract

The authors consider the following impulsive differential equations involving the one-dimensional singular p-Laplacian: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11">View MathML</a> are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11">View MathML</a>. In this case, our results cover equations without impulsive effects and are compared with some recent results.

Keywords:
multi-parameter; impulsive differential equations; one-dimensional singular p-Laplacian; positive solution; cone and partial ordering

1 Introduction

The theory and applications of the fourth-order ordinary differential equation are emerging as an important area of investigation; it is often referred to as the beam equation. In [1], Sun and Wang pointed out that it is necessary and important to consider various fourth-order boundary value problems (BVPs for short) according to different forms of supporting. Owing to its importance in engineering, physics, and material mechanics, fourth-order BVPs have attracted much attention from many authors; see, for example [2-29] and the references therein.

Very recently, Zhang and Liu [30] studied the following fourth-order four-point boundary value problem without impulsive effect:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M16">View MathML</a>. By using the upper and lower solution method, fixed point theorems, and the properties of the Green’s function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M18">View MathML</a>, the authors give sufficient conditions for the existence of one positive solution.

In this paper, we investigate the existence of positive solutions of fourth-order impulsive differential equations with two parameters

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M19">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11">View MathML</a> are two parameters, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M24">View MathML</a> is a p-Laplace operator, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M28">View MathML</a>, ω is a nonnegative measurable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M30">View MathML</a> on any open subinterval in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M29">View MathML</a> which may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M32">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M34">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>) (where m is fixed positive integer) are fixed points with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M37">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M39">View MathML</a> represent the right-hand limit and left-hand limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M40">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M41">View MathML</a>, respectively. In addition, ω, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M42">View MathML</a>, g, and h satisfy

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M43">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M44">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45">View MathML</a> for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M47">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M49">View MathML</a>) for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M51">View MathML</a> are nonnegative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M53">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M54">View MathML</a>

(1.2)

Some special cases of (1.1) have been investigated. For example, Bai and Wang [14] studied the existence of multiple solutions of problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M58">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>. By using a fixed point theorem and degree theory, the authors proved the existence of one or two positive solutions of problem (1.1).

Feng [31] considered problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M64">View MathML</a>. By using a suitably constructed cone and fixed point theory for cones, the author proved the existence results of multiple positive solutions of problem (1.1).

Motivated by the papers mentioned above, we will extend the results of [14,30,31] to problem (1.1). We remark that on impulsive differential equations with a parameter only a few results have been obtained, not to mention impulsive differential equations with two parameters; see, for instance, [32-34]. However, these results only dealt with the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M64">View MathML</a>.

The rest of the paper is organized as follows: in Section 2, we state the main results of problem (1.1). In Section 3, we provide some preliminary results, and the proofs of the main results together with several technical lemmas are given in Section 4.

2 Main results

In this section, we state the main results, including existence and multiplicity of positive solutions for problem (1.1).

We begin by introducing the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M67">View MathML</a>

We also choose four numbers r, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69">View MathML</a>, and R satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M70">View MathML</a>

(2.1)

where δ is defined in (3.20).

Theorem 2.1Assume that (H1)-(H4) hold.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M72">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76">View MathML</a>, problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M79">View MathML</a>

(2.2)

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M81">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M84">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76">View MathML</a>, problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M87">View MathML</a>

(2.3)

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M88">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M84">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M76">View MathML</a>, problem (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M93">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M94">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M95">View MathML</a>

(2.4)

Theorem 2.2Assume that (H1)-(H4) hold.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M97">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101">View MathML</a>, problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>with property (2.2).

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M104">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M105">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M106">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M108">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101">View MathML</a>, problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>with property (2.3).

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M112">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99">View MathML</a>such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101">View MathML</a>, problem (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M93">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M94">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M119">View MathML</a>

(2.5)

3 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M120">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M121">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122">View MathML</a> is a real Banach space with norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M123">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M125">View MathML</a>.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M126">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M127">View MathML</a> is called a solution of problem (1.1) if it satisfies (1.1).

We shall reduce problem (1.1) to an integral equation. To this goal, firstly by means of the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M128">View MathML</a>

(3.2)

we convert problem (1.1) into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M129">View MathML</a>

(3.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M130">View MathML</a>

(3.4)

Lemma 3.1If (H1), (H2), and (H4) hold, then problem (3.3) has a unique solutionxgiven by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M131">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M132">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M133">View MathML</a>

(3.7)

Proof The proof of Lemma 3.1 is similar to that of Lemma 2.1 in [31]. □

Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M134">View MathML</a>. Then from (3.6) and (3.7), we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M17">View MathML</a> have the following properties.

Proposition 3.1If (H4) holds, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M137">View MathML</a>

(3.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M138">View MathML</a>

(3.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M139">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M140">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M141">View MathML</a>

(3.12)

Remark 3.1 From (3.6) and (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M142">View MathML</a>

Lemma 3.2If (H1), (H3), and (H4) hold, then problem (3.4) has a unique solutionyandycan be expressed in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M143">View MathML</a>

(3.13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M144">View MathML</a>

(3.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M145">View MathML</a>

(3.15)

Proof The proof of Lemma 3.2 is similar to that of Lemma 2.2 in [31]. □

From (3.14) and (3.15), we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M147">View MathML</a> have the following properties.

Proposition 3.2If (H4) holds, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M148">View MathML</a>

(3.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M149">View MathML</a>

(3.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M150">View MathML</a>

(3.18)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M151">View MathML</a>

Suppose that y is a solution of problem (1.1). Then from Lemma 3.1 and Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M152">View MathML</a>

Define a cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M154">View MathML</a>

(3.19)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M155">View MathML</a>

(3.20)

It is easy to see K is a closed convex cone of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M122">View MathML</a>.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M157">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M158">View MathML</a>

(3.21)

From (3.21), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M159">View MathML</a> is a solution of problem (1.1) if and only if y is a fixed point of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a>.

Lemma 3.3Suppose that (H1)-(H4) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M161">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M162">View MathML</a>is completely continuous.

Proof The proof of Lemma 3.3 is similar to that of Lemma 2.4 in [31]. □

To obtain positive solutions of problem (1.1), the following fixed point theorem in cones is fundamental, which can be found in [[35], p.94].

Lemma 3.4LetPbe a cone in a real Banach spaceE. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M164">View MathML</a>are bounded open sets inEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M166">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M167">View MathML</a>

is completely continuous such that either

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M169">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M171">View MathML</a>, or

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M169">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M171">View MathML</a>,

thenAhas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M176">View MathML</a>.

Remark 3.2 To make the reader clear what <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M179">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M180">View MathML</a> mean, we give typical examples of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M164">View MathML</a>, e.g.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M183">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M185">View MathML</a>.

4 Proofs of the main results

For convenience we introduce the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M186">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M187">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188">View MathML</a> is a constant.

Proof of Theorem 2.1 Part (i). Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>) for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M193">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M195">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M196">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M199">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M200">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M201">View MathML</a>

(4.1)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M72">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M205">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M206">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M207">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M208">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M209">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M210">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M211">View MathML</a>

(4.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M212">View MathML</a>. Thus, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M213">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M214">View MathML</a>

and then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M215">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M216">View MathML</a>

(4.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M217">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M218">View MathML</a>

It follows from (4.4) and (4.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M219">View MathML</a>

(4.6)

Applying (b) of Lemma 3.4 to (4.1) and (4.6) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M221">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M222">View MathML</a>. Hence, since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225">View MathML</a>, it follows that (2.2) holds. This gives the proof of part (i).

Part (ii). Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>) for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M230">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M231">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M232">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M233">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M199">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M237">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M238">View MathML</a>

(4.7)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M81">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M205">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M244">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M210">View MathML</a> satisfy (4.2) and (4.3), respectively.

Similar to the proof of (4.6), we can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M247">View MathML</a>

(4.8)

Applying (a) of Lemma 3.4 to (4.7) and (4.8) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M249">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M250">View MathML</a>. Hence, since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>, it follows that (2.3) holds. This gives the proof of part (ii).

Consider part (iii). Choose two numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69">View MathML</a> satisfying (2.1). By part (i) and part (ii), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M258">View MathML</a>

(4.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M88">View MathML</a>, from the proof of part (i) and part (ii), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M260">View MathML</a>

(4.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M261">View MathML</a>

(4.11)

Applying Lemma 3.4 to (4.9)-(4.11) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M264">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M266">View MathML</a>. These are the desired distinct positive solutions of problem (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M74">View MathML</a> satisfying (2.4). Then the result of part (iii) follows. □

Proof of Theorem 2.2 Part (i). Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>) for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M273">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M188">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M275">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M276">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M279">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M280">View MathML</a>

(4.12)

Similar to the proof of (4.5), we can prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M281">View MathML</a>

(4.13)

It follows from (4.12) and (4.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M282">View MathML</a>

(4.14)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M284">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M286">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M206">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M288">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M289">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M290">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M291">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M292">View MathML</a>

(4.16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M212">View MathML</a>. Thus, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M213">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M295">View MathML</a>

and then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M296">View MathML</a>

This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M297">View MathML</a>

(4.17)

Applying (b) of Lemma 3.4 to (4.14) and (4.17) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M221">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M222">View MathML</a>. Hence, since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225">View MathML</a>, it follows that (2.2) holds. This gives the proof of part (i).

Part (ii). Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M4">View MathML</a>) for all t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M46">View MathML</a>, we can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M308">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M231">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M310">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M311">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M279">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M315">View MathML</a>

(4.18)

Similar to the proof of (4.5), we can prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M316">View MathML</a>

(4.19)

It follows from (4.18) and (4.19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M317">View MathML</a>

(4.20)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M319">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M286">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M184">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M323">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M291">View MathML</a> satisfy (4.15) and (4.16), respectively.

Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M277">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M327">View MathML</a>

This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M328">View MathML</a>

(4.21)

Applying (a) of Lemma 3.4 to (4.20) and (4.21) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M249">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M250">View MathML</a>. Hence, since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M223">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M225">View MathML</a>, it follows that (2.3) holds. This gives the proof of part (ii).

Consider part (iii). Choose two numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M69">View MathML</a> satisfying (2.1). By part (i) and part (ii), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M99">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M339">View MathML</a>

(4.22)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M340">View MathML</a>, from the proof of part (i) and part (ii), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M341">View MathML</a>

(4.23)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M342">View MathML</a>

(4.24)

Applying Lemma 3.4 to (4.22)-(4.24) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M160">View MathML</a> has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M264">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M266">View MathML</a>. These are the desired distinct positive solutions of problem (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M101">View MathML</a> satisfying (2.5). Then the proof of part (iii) is complete. □

Remark 4.1 Comparing with Feng [31], the main features of this paper are as follows.

(i) Two parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M11">View MathML</a> are considered.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M43">View MathML</a>, not only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M353">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M2">View MathML</a>.

(iii) It follows from the proof of Theorem 2.1 that the conditions of Corollary 3.2 in [31] are not the optimal conditions, which guarantee the existence of at least one positive solution for problem (1.1). In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M318">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/112/mathml/M357">View MathML</a>, we can prove that problem (1.1) has at least one positive solution, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

This work is sponsored by the project NSFC (11301178, 11171032) and the Fundamental Research Funds for the Central Universities (2014MS58). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.

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