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Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems

Chengbo Zhai* and Mengru Hao

Author Affiliations

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, P.R. China

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Boundary Value Problems 2013, 2013:85  doi:10.1186/1687-2770-2013-85

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/85


Received:22 November 2012
Accepted:18 March 2013
Published:10 April 2013

© 2013 Zhai and Hao; 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

This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order ν, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4">View MathML</a>. Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.

MSC: 26A33, 34B18, 34B27.

Keywords:
Riemann-Liouville fractional derivative; fractional differential equation; positive solution; existence and uniqueness; fixed point theorem for mixed monotone operator

1 Introduction

In this paper, we investigate the existence and uniqueness of positive solutions for the fractional boundary value problem (FBVP for short) of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M5">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M2">View MathML</a> is the standard Riemann-Liouville fractional derivative of order ν, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4">View MathML</a>.

Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc.; see [1-6] for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross [3], Podlubny [5], Kilbas et al.[6], and the papers [7-16] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10,14,17] for example.

In [18], Goodrich utilized the Krasnoselskii’s fixed point theorem to study a FBVP of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M9">View MathML</a>

(1.2)

and established the existence of at least one positive solution for FBVP (1.2). By using the same fixed point theorem, Goodrich [19] considered the existence of a positive solution to the following systems of differential equations of fractional order:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M12">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M14">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M15">View MathML</a>, with the following boundary value conditions:

under the assumptions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M18">View MathML</a>, f, g are nonnegative and continuous. But the uniqueness of positive solutions is not treated in these papers.

Different from the works mentioned above, motivated by the work [20], we will use two fixed point theorems for mixed monotone operators with perturbation to show the existence and uniqueness of positive solutions for FBVP (1.1). To our knowledge, there are still very few to utilize the fixed point results on mixed monotone operators with perturbation to study the existence and uniqueness of a positive solution for nonlinear fractional differential equation boundary value problems. So, it is worthwhile to investigate FBVP (1.1) by using our new fixed point theorems in [20]. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of fractional calculus and some definitions, notations and results of mixed monotone operators. In Section 3 we provide some conditions, under which the problem FBVP (1.1) has a unique positive solution. Finally, in Section 4, we provide an example, which explicates the applicability of our result.

2 Preliminaries

For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.

Definition 2.1 (See [18])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M19">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M20">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M21">View MathML</a>. Then the νth Riemann-Liouville fractional integral is defined to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M22">View MathML</a>

whenever the right-hand side is defined. Similarly, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M20">View MathML</a>, we define the νth Riemann-Liouville fractional derivative to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M25">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M3">View MathML</a> is the unique positive integer satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M28">View MathML</a>.

Lemma 2.2 (See [19])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M29">View MathML</a>be given. Then the unique solution to problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M30">View MathML</a>together with the boundary conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M31">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M33">View MathML</a>, is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M34">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M35">View MathML</a>

(2.2)

is the Green function for this problem.

Lemma 2.3 (See [19])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>be as given in the statement of Lemma 2.2. Then we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>is a continuous function on the unit square<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M38">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M39">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M40">View MathML</a>.

Lemma 2.4The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>defined by (2.2) satisfies the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M42">View MathML</a>

Proof Evidently, the right inequality holds. So, we only need to prove the left inequality. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M43">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M44">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M45">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M46">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M47">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M48">View MathML</a>

So, the proof is complete. □

In the sequel, we present some basic concepts in ordered Banach spaces for completeness and two fixed point theorems which we will be used later. For convenience of readers, we suggest that one refers to [20-22] for details.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M49">View MathML</a> is a real Banach space which is partially ordered by a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M50">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M52">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M54">View MathML</a>, then we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M55">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M56">View MathML</a>. By θ we denote the zero element of E. Recall that a non-empty closed convex set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M50">View MathML</a> is a cone if it satisfies (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M59">View MathML</a>; (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M61">View MathML</a>.

P is called normal if there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M62">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M64">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M65">View MathML</a>; in this case, N is called the normality constant of P. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M66">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M67">View MathML</a> is called the order interval between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M69">View MathML</a>. We say that an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M70">View MathML</a> is increasing (decreasing) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M51">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M72">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M73">View MathML</a>).

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M63">View MathML</a>, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M75">View MathML</a> means that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M77">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M78">View MathML</a>. Clearly, ∼ is an equivalence relation. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M81">View MathML</a>), we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a> the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M83">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M84">View MathML</a>.

Definition 2.5 (See [20,22])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85">View MathML</a> is said to be a mixed monotone operator if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M86">View MathML</a> is increasing in x and decreasing in y, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M89">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M90">View MathML</a>. Element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M58">View MathML</a> is called a fixed point of A if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M92">View MathML</a>.

Definition 2.6 An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M93">View MathML</a> is said to be sub-homogeneous if it is satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M94">View MathML</a>

(2.3)

Definition 2.7 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M95">View MathML</a> and β be a real number with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M96">View MathML</a>. An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M97">View MathML</a> is said to be β-concave if it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M98">View MathML</a>

(2.4)

Lemma 2.8 (See Theorem 2.1 in [20])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M100">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85">View MathML</a>is a mixed monotone operator and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M102">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M103">View MathML</a>is an increasing sub-homogeneous operator. Assume that

(i) there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M104">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M105">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M106">View MathML</a>;

(ii) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M109">View MathML</a>.

Then:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M111">View MathML</a>;

(2) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M114">View MathML</a>

(3) the operator equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M115">View MathML</a>has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M116">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>;

(4) for any initial values<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118">View MathML</a>, constructing successively the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M119">View MathML</a>

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M120">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M121">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122">View MathML</a>.

Lemma 2.9 (See Theorem 2.4 in [20])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M100">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M85">View MathML</a>is a mixed monotone operator and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M126">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M103">View MathML</a>is an increasingβ-concave operator. Assume that

(i) there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M104">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M105">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M106">View MathML</a>;

(ii) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M109">View MathML</a>.

Then:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M111">View MathML</a>;

(2) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M138">View MathML</a>

(3) the operator equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M115">View MathML</a>has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M116">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>;

(4) for any initial values<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118">View MathML</a>, constructing successively the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M143">View MathML</a>

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M120">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M145">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M146">View MathML</a>.

Remark 2.10 (i) If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M147">View MathML</a> in Lemma 2.8, then the corresponding conclusion is still true (see Corollary 2.2 in [20]); (ii) if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M148">View MathML</a> in Lemma 2.9, then the conclusion obtained is also true (see Theorem 2.7 in [23]).

3 Main results

In this section, we apply Lemma 2.8 and Lemma 2.9 to study FBVP (1.1), and we obtain some new results on the existence and uniqueness of positive solutions. The method used here is relatively new to the literature and so are the existence and uniqueness results to the fractional differential equations.

In our considerations, we work in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M149">View MathML</a> with the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M150">View MathML</a>. Notice that this space can be equipped with a partial order given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M151">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M152">View MathML</a>, the standard cone. It is clear that P is a normal cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M153">View MathML</a> and the normality constant is 1.

Theorem 3.1Assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M154">View MathML</a>is continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M155">View MathML</a>is continuous;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M156">View MathML</a>is increasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157">View MathML</a>for fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M158">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159">View MathML</a>, decreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159">View MathML</a>for fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M163">View MathML</a>is increasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157">View MathML</a>for fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M167">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162">View MathML</a>, and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M171">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M175">View MathML</a>;

(H4) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M107">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M179">View MathML</a>.

Then:

(1) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>and

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>is given as in (2.2);

(2) FBVP (1.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>;

(3) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, constructing successively the sequences

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>.

Proof To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M194">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a> is given as in (2.2).

Define two operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M197">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M198">View MathML</a>

It is easy to prove that u is the solution of FBVP (1.1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M199">View MathML</a>. From (H1), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M201">View MathML</a>. In the sequel, we check that A, B satisfy all the assumptions of Lemma 2.8.

Firstly, we prove that A is a mixed monotone operator. In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M203">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M205">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, and by (H2) and Lemma 2.3,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M209">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M210">View MathML</a>.

Further, it follows from (H2) and Lemma 2.3 that B is increasing. Next we show that A satisfies the condition (2.5). For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212">View MathML</a>, by (H3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M213">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M214">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212">View MathML</a>. So, the operator A satisfies (2.5). Also, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M218">View MathML</a>, from (H3) we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M219">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M220">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M218">View MathML</a>. That is, the operator B is sub-homogeneous. Now we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M224">View MathML</a>. On the one hand, from (H1), (H2) and Lemma 2.4, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M226">View MathML</a>

On the other hand, also from (H1), (H2) and Lemma 2.4, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M228">View MathML</a>

From (H2), (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M229">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M231">View MathML</a>

and in consequence,

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>; and hence we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223">View MathML</a>. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M236">View MathML</a>

from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166">View MathML</a>, we easily prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M238">View MathML</a>. Hence the condition (i) of Lemma 2.8 is satisfied.

In the following, we show the condition (ii) of Lemma 2.8 is satisfied. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M239">View MathML</a>, and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, from (H4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M241">View MathML</a>

Then we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M242">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212">View MathML</a>. Finally, an application of Lemma 2.8 implies: there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M247">View MathML</a>; the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M248">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; for any initial values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118">View MathML</a>, constructing successively the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M252">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M254">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122">View MathML</a>. That is,

FBVP (1.1) has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, the sequences

satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>. □

Theorem 3.2Assume (H1), (H2) and

(H5) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M171">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M162">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M269">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M175">View MathML</a>;

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M273">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M275">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M179">View MathML</a>.

Then:

(1) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>and

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>is given as in (2.2);

(2) FBVP (1.1) has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>;

(3) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, constructing successively the sequences

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>.

Sketch of the proof Consider two operators A, B defined in the proof of Theorem 3.1. Similarly, from (H1), (H2), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M200">View MathML</a> is a mixed monotone operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M201">View MathML</a> is increasing. From (H5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M295">View MathML</a>

From (H2), (H6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M296">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M273">View MathML</a>, we get

and in consequence,

So, we can easily prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M224">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M239">View MathML</a>, and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, from (H6),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M304">View MathML</a>

Then we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M305">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M212">View MathML</a>. Finally, an application of Lemma 2.9 implies: there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M247">View MathML</a>; the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M248">View MathML</a> has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; for any initial values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M118">View MathML</a>, constructing successively the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M315">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M254">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M122">View MathML</a>. That is,

FBVP (1.1) has a unique positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, the sequences

satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M325">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>. □

From Remark 2.10 and similar to the proofs of Theorems 3.1-3.2, we can prove the following conclusions.

Corollary 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M327">View MathML</a>. Assume thatfsatisfies the conditions of Theorem 3.1 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M328">View MathML</a>. Then: (i) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>and

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>is given as in (2.2); (ii) the FBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M336">View MathML</a>

has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; (iii) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, constructing successively the sequences

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>.

Corollary 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M344">View MathML</a>. Assume thatgsatisfies the conditions of Theorem 3.2 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M166">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>. Then: (i) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M182">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M350">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M36">View MathML</a>is given as in (2.2); (ii) the FBVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M354">View MathML</a>

has a unique positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M187">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>; (iii) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M189">View MathML</a>, constructing successively the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M358">View MathML</a>

we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M359">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M192">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M193">View MathML</a>.

4 An example

We now present one example to illustrate Theorem 3.1.

Example 4.1

Consider the following FBVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M362">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M363">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M364">View MathML</a> are continuous with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M365">View MathML</a>.

Obviously, problem (4.1) fits the framework of FBVP (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M367">View MathML</a>. (Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M368">View MathML</a>, therefore, in this case.) In this example, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M369">View MathML</a> and let

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M371">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M372">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M373">View MathML</a> is continuous with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M374">View MathML</a>. And <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M156">View MathML</a> is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M376">View MathML</a> for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159">View MathML</a>, decreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M159">View MathML</a> for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M381">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M163">View MathML</a> is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M157">View MathML</a> for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M161">View MathML</a>. Besides, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M387">View MathML</a>, we have

Moreover, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M389">View MathML</a>, then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M390">View MathML</a>

Hence all the conditions of Theorem 3.1 are satisfied. An application of Theorem 3.1 implies that problem (4.1) has a unique positive solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M82">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M392">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/85/mathml/M178">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

The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).

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