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Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter

Wenxia Wang*, Yanping Zheng, Hui Yang and Junxia Wang

Author Affiliations

Department of Mathematics, Taiyuan Normal University, Taiyuan, 030012, P.R. China

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Boundary Value Problems 2014, 2014:80  doi:10.1186/1687-2770-2014-80

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/80


Received:7 October 2013
Accepted:26 March 2014
Published:9 April 2014

© 2014 Wang et al.; licensee Springer.

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

Abstract

This paper is concerned with the existence, nonexistence, and uniqueness of convex monotone positive solutions of elastic beam equations with a parameter λ. The boundary conditions mean that the beam is fixed at one end and attached to a bearing device or freed at the other end. By using fixed point theorem of cone expansion, we show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1">View MathML</a> such that the beam equation has at least two, one, and no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M4">View MathML</a>, respectively; furthermore, by using cone theory we establish some uniqueness criteria for positive solutions for the beam and show that such solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> depends continuously on the parameter λ. In particular, we give an estimate for critical value of parameter λ.

MSC: 34B18, 34B15.

Keywords:
elastic beam equation; positive solution; fixed point; cone

1 Introduction and preliminaries

In this paper, we consider the following nonlinear fourth-order two-point boundary value problem (BVP) for elastic beam equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M6">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a> is a parameter. Throughout this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M10">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M11">View MathML</a> is called a positive solution of BVP (1) if x is a solution of BVP (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M13">View MathML</a>. A convex monotone positive solution means convex nondecreasing positive solution.

Because of characterization of the deformation of the equilibrium state, fourth-order boundary value problems for elastic beam equations are extensively applied to mechanics and engineering; see [1-3]. Some nonlinear elastic beam equations have been studied extensively. For a small sample of such work, we refer the reader to the work of Bai and Wang [4], Bai [5], Bonanno and Bellaa [6], Li [7], Liu and Li [8], Liu [9], Ma and Xu [10], and Ma and Thompson [11] on an elastic beam whose two ends are simply supported, the works of Yang [12] and Zhang [13] on an elastic beam of which one end is embedded and another end is fastened with a sliding clamp, and the work of Graef et al.[14] on multipoint boundary value problems.

BVP (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M14">View MathML</a> is called a cantilever beam equation, it describes the deflection of the elastic beam fixed at the left end and free at the right end. Existence and multiplicity of positive solutions of cantilever beam problems without parameter have been studied by some authors; see Yao [15,16] and references therein. BVP (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M15">View MathML</a> describes the deflection of the elastic beam fixed at the left end and attached to a bearing device given by the function −q at the right end. When the elastic beam equation does not contain parameter λ, the existence of multiple positive solutions and unique positive solution was presented in [17] by variational methods and in [18] by a fixed point theorem, respectively; monotone positive solutions were obtained by using the monotone iteration method in [19]. However, there are few papers concerned with positive solutions for BVP (1) with parameter, especially with the solution’s dependence on parameter λ in the existing literature. The aim of this paper is to show that the existence and number of convex monotone positive solutions of BVP (1) are affected by the parameter λ.

The paper is organized as follows. In Section 2, we present that a nontrivial and nonnegative solution of BVP (1) is convex monotone positive solution. In Section 3, we obtain some results on the existence, multiplicity and nonexistence of positive solutions for BVP (1). These results show that the number of positive solutions for BVP (1) depends on the parameter λ. In Section 4, we establish some uniqueness criteria for positive solutions for BVP (1) and show that such a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> depends continuously on the parameter λ. In particular, we give an estimate for the critical value of the parameter λ.

In the rest of this section, we introduce some notations and known results. For the reader’s convenience, we suggest that one refer to [20-22], and [23] for details.

Let E be a real Banach space and θ denote the zero element of E. A nonempty closed convex set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M17">View MathML</a> is called a cone of E if it satisfies (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M19">View MathML</a>; (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M21">View MathML</a>. E is partially ordered by the cone P, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M22">View MathML</a> iff <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M23">View MathML</a>. A cone P is said to be normal if there exists a positive number N, called the normal constant of P, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M24">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M25">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M27">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M28">View MathML</a>.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M29">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M30">View MathML</a> means that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M32">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M33">View MathML</a>. Clearly, ∼ is an equivalence relation. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M34">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M36">View MathML</a>), we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a> the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M38">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M39">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M40">View MathML</a>. An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M41">View MathML</a> is said to be increasing if for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M43">View MathML</a>. An element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M44">View MathML</a> is called a fixed point of T if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M45">View MathML</a>.

Lemma 1.1 (Fixed point theorem of cone expansion) [21,22]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M47">View MathML</a>are bounded open subsets ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M48">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M49">View MathML</a>be a completely continuous operator such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M50">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M52">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M53">View MathML</a>. ThenThas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M54">View MathML</a>.

Lemma 1.2[23]

LetPbe a normal cone inE, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M55">View MathML</a>be increasing and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M57">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M58">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M59">View MathML</a>. ThenThas a unique fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>. Moreover, constructing successively the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M62">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M65">View MathML</a>.

2 Solutions

In what follows, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M66">View MathML</a>, the Banach space of all continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M67">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M68">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M69">View MathML</a>. It is clear that P is a normal cone and its normality constant is 1.

From [18] and [19], it is evident that BVP (1) has an integral formulation given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M70">View MathML</a>

(2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M71">View MathML</a>

(3)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M72">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M73">View MathML</a>

(4)

Define operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M74">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M75">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M77">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M78">View MathML</a>.

It is clear from (2) that solving BVP (1) is equivalent to finding fixed points of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>. In particular, x is a fixed point of B iff x is a solution of the following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M80">View MathML</a>

and x is a fixed point of λA iff x is a solution of the following cantilever beam problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M81">View MathML</a>

(5)

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M82">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M83">View MathML</a>

(6)

then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M84">View MathML</a>is nondecreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M84">View MathML</a>is a convex function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M67">View MathML</a>.

Proof From (6), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M92">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M93">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M94">View MathML</a>. Thus, we complete the proof of the lemma. □

Now, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M95">View MathML</a>

then, it is easy to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M96">View MathML</a> is also a cone in E, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M97">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M98">View MathML</a>.

Lemma 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M101">View MathML</a>.

Proof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M103">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M105">View MathML</a>. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M18">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M107">View MathML</a>

By Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M108">View MathML</a> is convex and nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>. From (3) and (4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M110">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M111">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>. Thus, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M99">View MathML</a>. From the above proof, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M101">View MathML</a>. This ends the proof. □

Lemma 2.3

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M116">View MathML</a>is a completely continuous operator;

(ii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117">View MathML</a>is nondecreasing, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M118">View MathML</a>is a completely continuous operator.

Proof Similarly to the proof of Theorem 1 in [19], applying the Arzela-Ascoli Theorem, the proof can be completed. □

From the proof of Lemma 2.2 we can show the following result.

Theorem 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M119">View MathML</a>is a solution for BVP (1), thenxis a convex monotone positive solution for BVP (1).

So, in the following sections, we only need to study solutions for BVP (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120">View MathML</a>.

3 Existence and nonexistence results

It is obvious from Lemma 2.2 that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M119">View MathML</a> is a solution for BVP (1) then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M122">View MathML</a>. So in this section, we will apply Lemma 1.1 to study the existence, multiplicity and nonexistence of solutions for BVP (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M123">View MathML</a>. It is reasonable that the domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> is restricted on K. The following conditions will be assumed:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M125">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a> for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M127">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M130">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M131">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M132">View MathML</a>;

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M133">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M134">View MathML</a>

(7)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M135">View MathML</a>.

Lemma 3.1Suppose that (H1)-(H3) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M136">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M137">View MathML</a>.

Proof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M136">View MathML</a> means that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M140">View MathML</a>. Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M141">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M142">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a> . From (H1) and (H2) we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M146">View MathML</a>. By Lemma 2.3 and (H3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M147">View MathML</a> converges to a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M137">View MathML</a>. This completes the proof. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M154">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M155">View MathML</a>

Theorem 3.2Suppose that (H1)-(H3) hold.

(i) If (H4) holds, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>has minimal and maximal fixed points in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158">View MathML</a>. Moreover, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>has at least one and has no fixed points in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165">View MathML</a>, then when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M166">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M167">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>has at least one and no fixed points in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M169">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively; when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>has at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M149">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M175">View MathML</a>.

Proof (i) From (H1), (H3), and (H4) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M176">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M178">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a> , then from (H1) and (H2) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M182">View MathML</a>

(8)

Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M184">View MathML</a> converge to fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M186">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>, respectively. From (8) it is evident that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M188">View MathML</a> are the minimal fixed point and maximal fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157">View MathML</a>, respectively. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a> we can complete the rest of the proof.

(ii) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M192">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M193">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M195">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M199">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M200">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M201">View MathML</a>

Similarly to the proof of Lemma 3.1, we can show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M202">View MathML</a>. The conclusion (ii) follows from Lemma 3.1 and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a>. This completes the proof of Theorem 3.2. □

Lemma 3.3Suppose that (H1)-(H3) hold and that one of (H5) and (H6) holds. If Λ is nonempty, then

(i) Λ is bounded from above, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205">View MathML</a>.

Proof (i) Suppose to the contrary that there exists an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M206">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M207">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M208">View MathML</a> is a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M209">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M210">View MathML</a>. There are two cases to be considered.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a> is bounded, that is, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M212">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M213">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M214">View MathML</a> . Hence, from (H1), (H3), and (4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M215">View MathML</a>

which is a contradiction.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a> is unbounded, that is, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M219">View MathML</a>.

When (H5) holds, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M220">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M222">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M225">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M226">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224">View MathML</a>. Moreover, from (H1) and the definition of K, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M229">View MathML</a>

which is a contradiction.

When (H6) holds, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M230">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M231">View MathML</a>. There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M232">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M233">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M234">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M235">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M236">View MathML</a>, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M237">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M238">View MathML</a>

which is a contradiction.

Consequently, we find that Λ is bounded from above.

(ii) By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a>, there exists a nondecreasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M240">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M241">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M208">View MathML</a> be a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M209">View MathML</a>. Arguing similarly as above in case 2, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M244">View MathML</a> is a bounded subset in K, that is, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M212">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a> ; on the other hand, note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M248">View MathML</a>

we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a> is an equicontinuous subset in K. Consequently, by an application of the Arzela-Ascoli Theorem we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M211">View MathML</a> is a relatively compact set in K. So, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M251">View MathML</a> converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M252">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M253">View MathML</a>

By taking the limit we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M254">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205">View MathML</a>. The proof is complete. □

Theorem 3.4Suppose that (H1)-(H4) hold and that one of (H5) and (H6) holds. Then, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M1">View MathML</a>such that BVP (1) has at least two, one, and no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M3">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively.

Proof Theorem 3.2 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M260">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M261">View MathML</a>. From Lemmas 3.1 and 3.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M262">View MathML</a>. Therefore, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a> we only to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has at least two fixed points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M123">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M266">View MathML</a>.

Now, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158">View MathML</a>. Theorem 3.2 means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M269">View MathML</a> which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M270">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M271">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M272">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, so for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M97">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M275">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M276">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M277">View MathML</a>

(9)

When (H5) holds, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M278">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M279">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M280">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M224">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M283">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M284">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M285">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M286">View MathML</a>

When (H6) holds, from the proof of Lemma 3.3 we can set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M287">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M288">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M289">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M290">View MathML</a>.

Consequently, in virtue of Lemma 1.1 we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has another fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M292">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M293">View MathML</a>

Equation (9) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has no fixed points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M295">View MathML</a>. In conclusion, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has at least two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M292">View MathML</a> in K with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M300">View MathML</a>. The proof is complete. □

Remark 3.1 In the above results, we can replace (H5) with the following condition: there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M301">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M302">View MathML</a>.

In the following, we give some sufficient conditions that BVP (1) has no positive solutions.

Theorem 3.5Suppose that there exists a nonnegative integrable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M303">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M307">View MathML</a>. Then BVP (1) has no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M308">View MathML</a>.

Proof Assume to the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M309">View MathML</a> is a solution of BVP (1), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M310">View MathML</a>, which is a contradiction. The proof is complete. □

Similarly to the proof of Theorem 3.5, we can easily obtain the following results.

Theorem 3.6Suppose that there exist an integrable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M311">View MathML</a>and a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M312">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M317">View MathML</a>. Then BVP (1) has no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M318">View MathML</a>.

Theorem 3.7Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>. Then BVP (1) has no positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a>.

Remark 3.2 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M14">View MathML</a>, BVP (1) becomes a cantilever beam problem (5). In this case, we can delete the conditions on q in Theorems 3.2, 3.4-3.6 and obtain the following corresponding results for BVP (5).

Suppose that (H1) and (H3) hold. Then BVP (5) has minimal and maximal solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M324">View MathML</a>. Further, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M325">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M326">View MathML</a> such that BVP (5) has at least one and has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172">View MathML</a> then BVP (5) has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M175">View MathML</a>.

Suppose that (H1), (H3), and (H5) hold. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M331">View MathML</a> and BVP (5) has at least two, one and has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M332">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively.

Under the conditions in Theorem 3.5, BVP (5) has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M335">View MathML</a>.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M337">View MathML</a> satisfy the conditions in Theorem 3.6, then BVP (5) has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M338">View MathML</a>.

Remark 3.3 (i) We give an example to illustrate Theorem 3.2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M339">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M340">View MathML</a>

By straightforward calculations we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M345">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346">View MathML</a>. So the conditions in Theorem 3.2 are satisfied. Therefore, by Theorem 3.2 we find that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M347">View MathML</a> such that BVP (1) has minimal and maximal solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M157">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M349">View MathML</a>, has at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M350">View MathML</a> and has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M154">View MathML</a>.

We give another example to illustrate Theorem 3.4. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M354">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M355">View MathML</a>

A straightforward calculation can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M356">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M357">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M358">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M359">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M360">View MathML</a> Therefore, the conditions (H1)-(H5) hold. Thus, by Theorem 3.4 we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M361">View MathML</a> such that BVP (1) has at least two, one, and no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M363">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M163">View MathML</a>, respectively.

(ii) In Theorems 3.5-3.7, we do not require f and q to be monotone in x. For example, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M365">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M366">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M367">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M368">View MathML</a>, then the conditions in Theorem 3.6 are satisfied and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M369">View MathML</a>. So by Theorem 3.6 we find that BVP (1) has no positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M370">View MathML</a>.

4 Uniqueness and dependence on parameter

In this section, we will apply cone theory to further study the uniqueness of solution for BVP (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120">View MathML</a> and the dependence of such a positive solution on the parameter λ. The following hypotheses are needed:

(H7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M372">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M375">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M376">View MathML</a>;

(H8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M377">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>;

(H9) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M385">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M386">View MathML</a>.

Remark 4.1 The inequalities in (H7), (H8), and (H9) are equivalent to the following inequalities, respectively:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M387">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M388">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a> as in Section 1. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M39">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M392">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M394">View MathML</a>.

Remark 4.2 (H2) and (H7) imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M395">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M396">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M397">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>.

Remark 4.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> be a solution for BVP (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M120">View MathML</a>. If (H2) and (H7) hold, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M401">View MathML</a>. Indeed, from Theorem 2.4 we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M402">View MathML</a>. So Remark 4.2 implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M403">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M404">View MathML</a>

we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M401">View MathML</a>.

So, in this section, we only need to consider the unique solution for BVP (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>.

Lemma 4.1Assume that (H2) and (H7) hold. ThenBhas a unique fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>, moreover, constructing successively the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M409">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>) for any initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M412">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M414">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M415">View MathML</a>. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, from (H7) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M418">View MathML</a>. Consequently, the conclusion follows from Lemma 1.2. This completes the proof. □

Lemma 4.2Assume that (H1), (H2), (H7), and (H8) hold. Then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M419">View MathML</a>is an increasing operator;

(ii) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M422">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M423">View MathML</a>;

(iii) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M424">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M426">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M427">View MathML</a>

Proof The conclusion (i) follows from (H1), (H2), (H7), and (4).

The proof of (ii). For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M56">View MathML</a>, from (H1) and (4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M430">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M431">View MathML</a>

(10)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M432">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M433">View MathML</a>

(11)

The proof of (iii). For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M435">View MathML</a>, from (10) and (11) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M436">View MathML</a>

Moreover, from (H7) and (H8) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M437">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M438">View MathML</a>. This completes the proof. □

Lemma 4.3Assume that (H1), (H2), (H7), and (H8) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>has a unique fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>iff there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M442">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M443">View MathML</a>. Moreover, constructing successively the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M144">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>) for any initial value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M447">View MathML</a>

(12)

Proof ‘⇒’ Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> be a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M451">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M452">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M443">View MathML</a>.

‘⇐’ By virtue of Lemma 4.1, B has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M456">View MathML</a>

(13)

Now, we are going to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M457">View MathML</a>

(14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M458">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M459">View MathML</a>. Otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M460">View MathML</a>, from Lemma 4.2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M461">View MathML</a>

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M462">View MathML</a>, we get a contradiction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M463">View MathML</a>. Thus, (14) holds.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M464">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M465">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M214">View MathML</a> . From (13) and (14) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M468">View MathML</a>

(15)

Lemma 2.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M469">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M470">View MathML</a> converge to fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M472">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a>, respectively. From (15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M474">View MathML</a>

(16)

To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> has only one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M476">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M477">View MathML</a>

(17)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M478">View MathML</a>

(18)

From (16)-(18) we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M479">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M480">View MathML</a>. We assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M481">View MathML</a>. Otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M482">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M483">View MathML</a>, then by Lemma 4.2 we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M484">View MathML</a>

By (17), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M485">View MathML</a>, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M486">View MathML</a>, which is a contradiction. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M481">View MathML</a>. Thus, by (16) and (18) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M488">View MathML</a>

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M489">View MathML</a> is the unique fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M476">View MathML</a>.

Now, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M60">View MathML</a> is the unique fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>. By the above proof, we only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> does not have any fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M496">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M497">View MathML</a> is a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M496">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M500">View MathML</a>

(19)

It is evident that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M501">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M502">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M503">View MathML</a>. By Lemma 4.2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M504">View MathML</a>

Thus, from (19) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M505">View MathML</a>, which is a contradiction. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M506">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M507">View MathML</a>, which implies the contradiction: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M508">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M509">View MathML</a>.

Finally, the iterative scheme and (12) can be proved in a similar way to the proof of Theorem 3.4 of [21], here it is omitted. The proof is complete. □

Theorem 4.4Assume that (H1), (H2), (H7), and (H8) hold. Then there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M510">View MathML</a>such that BVP (1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>and does not have any solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515">View MathML</a>. Moreover, set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M516">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, then (12) holds.

Proof By Lemma 4.1, B has the unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M521">View MathML</a>, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M522">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M523">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M524">View MathML</a>

(20)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M525">View MathML</a>. Lemma 4.3 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M526">View MathML</a>

(21)

Similarly to the proof of Lemma 3.1, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M527">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M528">View MathML</a>.

Now, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M529">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M530">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M531">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M533">View MathML</a>. By (H7), (H8), and (20), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M534">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M535">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M536">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M537">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M538">View MathML</a>. We assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M539">View MathML</a>. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M540">View MathML</a>, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a> it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M539">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M205">View MathML</a>. Then by (14) and (21) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M545">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M546">View MathML</a>. Similarly to the proof of (20), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M547">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M548">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M549">View MathML</a>

(22)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M550">View MathML</a> for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M551">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M552">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M553">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M554">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M555">View MathML</a>. Therefore, from (22) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M556">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M557">View MathML</a>, which is a contradiction to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M559">View MathML</a>. Consequently, an application of Lemma 4.3 completes the proof. □

In what follows, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M407">View MathML</a> is the unique fixed point of B in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> is the unique fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M537">View MathML</a>.

Theorem 4.5Assume that (H1), (H2), (H7), and (H8) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>depends upon the parameterλas follows:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is nondecreasing with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is continuous with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M572">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M573">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M574">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M575">View MathML</a>, from the proof of Lemma 4.3, we find that the unique fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M576">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M577">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M578">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M579">View MathML</a>.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M580">View MathML</a>. In order to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M581">View MathML</a>, let sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M582">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M583">View MathML</a>

By virtue of the above conclusion (i) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M584">View MathML</a>

(23)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M585">View MathML</a> is a bounded subset in P. Further, similarly to the proof of the conclusion (ii) in Lemma 3.3 we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M585">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M587">View MathML</a>. From (23) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M588">View MathML</a>, which leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M589">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M590">View MathML</a>

By taking the limit we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M591">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M592">View MathML</a> has only one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M594">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M595">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M596">View MathML</a>.

A similar argument can show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M597">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M598">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M599">View MathML</a>. Thus, the proof of (ii) is complete.

(iii) It is obvious from the above conclusion (ii) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571">View MathML</a>.

In order to finish the proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M601">View MathML</a>, we consider two cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M540">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M603">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M604">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M572">View MathML</a>.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M204">View MathML</a>.

By the above conclusion (i) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M607">View MathML</a>. Suppose to the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M608">View MathML</a>. Similarly to the case 2 in the proof of Lemma 3.3, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M609">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M610">View MathML</a>. From Remark 4.3 we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M589">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M612">View MathML</a>, which is a contradiction. This ends the proof. □

Now, we give an estimate for critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a> in Theorem 4.4. If (H1) and (H8) hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M614">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M615">View MathML</a>.

Theorem 4.6Assume that (H1), (H2), (H7), and (H8) hold. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M616">View MathML</a>

(24)

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M230">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M618">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M619">View MathML</a>

(25)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M620">View MathML</a>, we can choose a sufficiently large positive integer number k such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M621">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M622">View MathML</a>

(26)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M623">View MathML</a>, then, from (4), (25), and (26) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M624">View MathML</a>

Moreover, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M625">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M626">View MathML</a>

Consequently, from (21) we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M627">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M628">View MathML</a>, which implies that (24) holds. This completes the proof. □

Remark 4.4 Different from Theorems 3.2 and 3.4, the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M191">View MathML</a> in Theorem 4.6 does not take into account effect of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M117">View MathML</a>. This is valuable, because the conditions (H2) and (H7) cannot ensure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M632">View MathML</a>. Certainly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M165">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M634">View MathML</a>. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M636">View MathML</a>.

Corollary 4.7Assume that (H1), (H2), (H7), and (H9) hold. Then

(i) BVP (1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M516">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is nondecreasing with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is continuous with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M649">View MathML</a>.

Proof From (H1), (H2), (H7), and (4), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M650">View MathML</a> is increasing for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a>. Further, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M653">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M654">View MathML</a>. Thus, the conclusion (i) follows from Lemma 1.2.

From (H9), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>. Therefore, in the same way as in the proof of Theorem 4.5, we can complete the rest of the proof. □

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M659">View MathML</a> is a constant function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M661">View MathML</a>. It is evident that B satisfies (H2) and (H7). So we can obtain the following two results.

Corollary 4.8Assume that (H1) and (H8) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M166">View MathML</a>, then

(i) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M663">View MathML</a>such that BVP (1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M664">View MathML</a>has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>and does not have any solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515">View MathML</a>. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M671">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is nondecreasing with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is continuous with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M513">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M679">View MathML</a>.

Corollary 4.9Assume that (H1) and (H8) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M172">View MathML</a>, then

(i) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>, BVP (1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M664">View MathML</a>has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M37">View MathML</a>, moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M686">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M643">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is nondecreasing inλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a>is continuous with respect toλfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M639">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M571">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M649">View MathML</a>.

Corollary 4.10Assume that (H1) and (H9) hold. Then the conclusions (i), (ii), (iii), and (iv) in Corollary 4.9 hold.

Finally, we give two concrete examples to illustrate those results in the section.

Example 1 In BVP (1), let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M695">View MathML</a>

it is obvious that the conditions (H1) and (H2) are satisfied. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>,

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M697">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M698">View MathML</a>;

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M699">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M700">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M701">View MathML</a>;

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M699">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M703">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M704">View MathML</a>,

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M378">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>. Similarly, we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M708">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M709">View MathML</a>. Therefore, the conditions (H7) and (H8) are satisfied. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M710">View MathML</a>

By Theorems 4.4, 4.5 and Remarks 4.3, 4.4 we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M711">View MathML</a> such that BVP (1) has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M713">View MathML</a> and does not have any positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M515">View MathML</a>. Moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M716">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M718">View MathML</a>, and such solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719">View MathML</a> satisfies the properties (i), (ii), and (iii) in Theorem 4.5.

Example 2 In BVP (1), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M720">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M721">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M126">View MathML</a>, it is easy to see that (H1) holds. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M374">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M725">View MathML</a>

So, (H8) holds. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M726">View MathML</a>

by Corollary 4.9 we find that BVP (1) has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M5">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M7">View MathML</a>. Moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M64">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M730">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M63">View MathML</a>), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M718">View MathML</a>, and such a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719">View MathML</a> satisfies the properties (ii), (iii), and (iv) in Corollary 4.9 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M734">View MathML</a>.

In this example, by using Wolfram Mathematica 9.0, we can plot the graphs of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M719">View MathML</a> for BVP (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M736','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M736">View MathML</a>, as the Figure 1 shows.

thumbnailFigure 1. Solutions for BVP (1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M737">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/80/mathml/M738">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments. This research was supported by the NNSF of China (11361047), the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021, 2013156), Research Project Supported by Shanxi Scholarship Council of China (2013-102) and the Science Foundation of Qinghai Province of China (2012-Z-910).

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