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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivatives

Yulin Zhao1*, Haibo Chen2 and Qiming Zhang1

Author Affiliations

1 School of Science, Hunan University of Technology, Zhuzhou, 412007, China

2 Department of Mathematics, Central South University, Changsha, 410075, China

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Boundary Value Problems 2013, 2013:103  doi:10.1186/1687-2770-2013-103

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/103


Received:17 January 2013
Accepted:12 April 2013
Published:25 April 2013

© 2013 Zhao et al.; licensee Springer

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

Abstract

In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given.

MSC: 34A08, 34B18, 39A13.

Keywords:
fractional q-difference equation; nonhomogeneous boundary value problem; positive solution; multiplicity

1 Introduction

Fractional differential equations have attracted considerable interest because of its demonstrated applications in various fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability [1,2]. Many researchers have studied the existence of solutions (or positive solutions) to fractional boundary value problems; for example, see [3-10] and the references therein.

The early work on q-difference calculus or quantum calculus dates back to Jackson’s papers [11], basic definitions and properties of quantum calculus can be found in the book [12]. For some recent existence results on q-difference equations, we refer to [13-15] and the references therein.

The fractional q-difference calculus had its origin in the works by Al-Salam [16] and Agarwal [17]. More recently, there seems to be new interest in the study of this subject and many new developments were made in this theory of fractional q-difference calculus [18-22]. Specifically, fractional q-difference equations have attracted the attentions of several researchers. Some recent work on the existence theory of fractional q-difference equations can be found in [20,23-31]. However, the study of boundary value problems for nonlinear fractional q-difference equations is still in the initial stage and many aspects of this topic need to be explored.

By using a fixed-point theorem in a cone, M. El-Shahed and F. Al-Askar [25] were concerned with the existence of positive solutions to nonlinear q-difference equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M3">View MathML</a> is the fractional q-derivatives of the Caputo type.

In [27], Graef and Kong investigated the boundary value problem with fractional q-derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M5">View MathML</a> is a parameter, and the uniqueness, existence and nonexistence of positive solutions are considered in terms of different ranges of λ.

By applying the Banach contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative, Ahmad, Ntouyas and Purnaras [29] studied the existence of solution for the following nonlinear fractional q-difference equation with nonlocal boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M3">View MathML</a> is the fractional q-derivative of the Caputo type, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M8">View MathML</a>.

Recently, in [32], the authors investigate the following singular semipositone integral boundary value problem for fractional q-derivatives equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M11">View MathML</a> is the q-derivative of Riemann-Liouville type of order α, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M12">View MathML</a> is continuous and semipositone, and may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M13">View MathML</a>.

Since finding positive solutions of boundary value problems is interest in various fields of sciences, fractional q-calculus equations has tremendous potential for applications. In this paper, we will deal with the following nonhomogeneous boundary value problem with fractional q-derivatives:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M14">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M18">View MathML</a>, and λ is a parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M11">View MathML</a> is the q-derivative of Riemann-Liouville type of order α, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M20">View MathML</a> is continuous. In the present work, we gave the corresponding Green’s function of the boundary value problem (1.1) and its properties. By using the generalized Banach contraction principle and Krasnoselskii’s fixed-point theorem, the uniqueness, existence, and multiplicity of positive solution to the BVP (1.1) are obtained in term of the explicit intervals for the nonhomogeneous term. Our results are different from those of [25,27].

2 Preliminaries on q-calculus and lemmas

For the convenience of the reader, below we cite some definitions and fundamental results on q-calculus as well as the fractional q-calculus. The presentation here can be found in, for example, [12,18,20,22].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M15">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M22">View MathML</a>

The q-analogue of the power function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M23">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M24">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M25">View MathML</a>

More generally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M26">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M27">View MathML</a>

(2.1)

Clearly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M28">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M29">View MathML</a>. The q-gamma function is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M30">View MathML</a>

and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M31">View MathML</a>.

The q-derivative of a function f is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M32">View MathML</a>

and q-derivatives of higher order by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M33">View MathML</a>

The q-integral of a function f defined in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M34">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M35">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M36">View MathML</a> and f is defined in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M34">View MathML</a>, then its integral from a to b is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M38">View MathML</a>

Similar to that for derivatives, an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M39">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M40">View MathML</a>

The fundamental theorem of calculus applies to these operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M42">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M43">View MathML</a>

and if f is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M44">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M45">View MathML</a>

(2.2)

The following formulas will be used later, namely, the integration by parts formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M46">View MathML</a>

and

(2.3)

(2.4)

(2.5)

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M51">View MathML</a> denotes the derivative with respect to the variable t.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52">View MathML</a> and f be a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M53">View MathML</a>. The fractional q-integral of Riemann-Liouville type is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M54">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M55">View MathML</a>

Definition 2.2 The fractional q-derivative of the Riemann-Liouville type of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M57">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M59">View MathML</a> is the smallest integer greater than or equal to α.

Lemma 2.3 ([20])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M61">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M62">View MathML</a>.

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M63">View MathML</a>andfbe a function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a>. Then the following formulas hold:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M65">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M66">View MathML</a>.

Lemma 2.5 ([20])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M67">View MathML</a>andnbe a positive integer. Then the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M68">View MathML</a>

Lemma 2.6 ([22])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M70">View MathML</a>, the following is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M71">View MathML</a>

Particularly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M73">View MathML</a>, using q-integration by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M74">View MathML</a>

Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M75">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M76">View MathML</a>

In order to define the solution for the problem (1.1), we need the following lemmas.

Lemma 2.7For given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M77">View MathML</a>, the unique solution of the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M78">View MathML</a>

(2.7)

subject to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M79">View MathML</a>

(2.8)

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M80">View MathML</a>

(2.9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M81">View MathML</a>

(2.10)

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M16">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M83">View MathML</a>. In view of Definition 2.1 and Lemma 2.4, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M84">View MathML</a>

Then it follows from Lemma 2.5 that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M85">View MathML</a> of (2.7) and (2.8) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M86">View MathML</a>

(2.11)

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M87">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M88">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M89">View MathML</a>.

Differentiating both sides of (2.11) and with the help of (2.4) and (2.6), we obtain,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M90">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M91">View MathML</a>

Then by the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M92">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M93">View MathML</a>. Using the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M94">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M95">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M96">View MathML</a>

This completes the proof of the lemma. □

Lemma 2.8The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M97">View MathML</a>defined by (2.10) satisfies the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M98">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M99">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M100">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M101">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M100">View MathML</a>.

Proof We start by defining the following two functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M103">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M104">View MathML</a>. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M105">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M106">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M107">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M98">View MathML</a>.

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M109">View MathML</a>, it follows from (2.4) and Lemma 2.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M110">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M111">View MathML</a> is an increasing function with respect to t. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M112">View MathML</a> is increasing in t. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M113">View MathML</a> is an increasing function of t for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M109">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M99">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M116">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M117">View MathML</a>

Finally, we prove part (ii). When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M118">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M119">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M116">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M121">View MathML</a>

which implies that part (ii) holds. This completes the proof of the lemma. □

Remark 2.9 If we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M122">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M123">View MathML</a>

According to [20], we may take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M125">View MathML</a>.

3 The main results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M126">View MathML</a> be a Banach space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M127">View MathML</a>. Define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M128">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M129">View MathML</a>.

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M130">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M131">View MathML</a>

(3.1)

Theorem 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M132">View MathML</a>is continuous and there exists a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M133">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M134">View MathML</a>

(3.2)

Then the BVP (1.1) has a unique positive solution for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M135">View MathML</a>, provided

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M136">View MathML</a>

(3.3)

If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a>, then the conclusion is true for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>.

Proof We will show that under the assumptions (3.2) and (3.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M140">View MathML</a> is a contraction operator for m sufficiently large.

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M113">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M142">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M143">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M144">View MathML</a>.

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M145">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M146">View MathML</a>.

By introduction, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M147">View MathML</a>

From the condition (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M148">View MathML</a>

for m sufficiently large. So, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M149">View MathML</a>

Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M135">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>, then the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M152">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a> and Lemma 2.8 imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M154">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M155">View MathML</a>. This completes the proof of the theorem. □

Remark 3.2 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M156">View MathML</a> is a constant, the condition (3.2) reduces to a Lipschitz condition.

Our next existence results is based on Krasnoselskii’s fixed-point theorem [33].

Lemma 3.3 (Krasnoselskii’s)

LetEbe a Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M157">View MathML</a>be a cone. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M159">View MathML</a>are open subsets ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M161">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M162">View MathML</a>be a completely continuous operator such that, either

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M166">View MathML</a>, or

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M164">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M166">View MathML</a>.

ThenThas at least one fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M171">View MathML</a>.

Define a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M172">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M173">View MathML</a>

Obviously, K is a cone of nonnegative functions in X.

Lemma 3.4The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M174">View MathML</a>is completely continuous.

Proof Firstly, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M175">View MathML</a>. By (2.9) and Lemma 2.8, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M176">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M177">View MathML</a>

Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M175">View MathML</a>.

Next, we show that T is uniformly bounded. For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M179">View MathML</a>, consider a bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M180">View MathML</a> of K defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M181">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M182">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M183">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M184">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M185">View MathML</a> is bounded.

Finally, we show that T is equicontinuous. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M186">View MathML</a>, setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M187">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M188">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M183">View MathML</a>, we can prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M191">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M192">View MathML</a>

In fact, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M193">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M194">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M195">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M197">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M198">View MathML</a>

By means of Arzela-Ascoli theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M174">View MathML</a> is completely continuous.

For the sake of convenience, we introduce the following weight functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M200">View MathML</a>

and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M201">View MathML</a>

 □

Theorem 3.5Suppose that there exists two positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M202">View MathML</a>such that one of the following conditions is satisfied

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M205">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M206">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M208">View MathML</a>.

Then the BVP (1.1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M209">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M210">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M211">View MathML</a>. If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a>, then the conclusion is true for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>.

Proof Because the proofs are similar, we prove only the case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203">View MathML</a>). Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M216">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M217">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M220">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M222">View MathML</a>. By assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M203">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M224">View MathML</a>

In view of (2.9) and Lemma 2.8, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M225">View MathML</a>

On the other hand, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M226">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M228">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M229">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M220">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M232">View MathML</a>

It follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M233">View MathML</a>

By Lemma 3.3, the operator T has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M234">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M210">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M237">View MathML</a>, then, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M238">View MathML</a> is positive for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M239">View MathML</a>. As in the proof of Theorem 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M240">View MathML</a> is a positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>. This completes the proof of the theorem. □

Theorem 3.6Suppose that there exists three positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M242">View MathML</a>such that one of the following conditions is satisfied

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M243">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M246">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M250">View MathML</a>.

Then the BVP (1.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M251">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M252">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M253">View MathML</a>. If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a>, then the conclusion is true for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>.

Proof We prove only the case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247">View MathML</a>). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M258">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M259">View MathML</a>, there exist two positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M261">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M264">View MathML</a>. Thus, it follows from the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M247">View MathML</a>) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M266">View MathML</a>

From Theorem 3.5, the operator T has two fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M268">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M269">View MathML</a>. Therefore, the BVP (1.1) has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M270">View MathML</a>. As in the proof of Theorem 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M271">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M272">View MathML</a> are two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>. This completes the proof of the theorem. □

Denote the integer part of m by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M274">View MathML</a>. Generally, we have the following theorem.

Theorem 3.7Suppose that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M275">View MathML</a>positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M276">View MathML</a>such that one of the following conditions is satisfied:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M277">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M280">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M281">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M280">View MathML</a>.

Then the BVP (1.1) has at leastmpositive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M286">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M287">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M211">View MathML</a>. If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M137">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M64">View MathML</a>, then the conclusion is true for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M139">View MathML</a>.

4 Examples

Example 4.1 The fractional q-difference boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M292">View MathML</a>

(4.1)

has a unique positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M293">View MathML</a>.

Proof In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M239">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M299">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M300">View MathML</a>. It is easy to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M301">View MathML</a>

A simple computation showed

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M302">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M303">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M304">View MathML</a>

Obviously, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M305">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M306">View MathML</a>

Thus, Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M293">View MathML</a>. □

Example 4.2 Consider the following fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M308">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M312">View MathML</a>. Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M313">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M314">View MathML</a>.

By calculation, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M315">View MathML</a>. By Lemma 2.6, Lemma 2.8 and with the aid of a computer, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M316">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M317">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M318">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M320">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M321">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M322">View MathML</a> satisfies

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M323">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M324">View MathML</a>.

So, by Theorem 3.5, the problem (4.2) has one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M238">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M326">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/103/mathml/M327">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (Grant No. 11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ3106, 12JJ2004), and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12B034).

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