SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales

Fatma Tokmak and Ilkay Yaslan Karaca*

Author affiliations

Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey

For all author emails, please log on.

Citation and License

Boundary Value Problems 2013, 2013:52  doi:10.1186/1687-2770-2013-52

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/52


Received:30 September 2012
Accepted:12 February 2013
Published:14 March 2013

© 2013 Tokmak and Karaca; licensee Springer

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

Abstract

In this paper, we make use of the four functionals fixed point theorem to verify the existence of at least one symmetric positive solution of a second-order m-point boundary value problem on time scales such that the considered equation admits a nonlinear term f whose sign is allowed to change. The discussed problem involves both an increasing homeomorphism and homomorphism, which generalizes the p-Laplacian operator. An example which supports our theoretical results is also indicated.

MSC: 34B10, 39A10.

Keywords:
symmetric positive solution; fixed-point theorem; time scales; m-point boundary value problem; increasing homeomorphism and homomorphism

1 Introduction

The theory of time scales was introduced by Stefan Hilger [1] in his PhD thesis in 1988 in order to unify continuous and discrete analysis. This theory was developed by Agarwal, Bohner, Peterson, Henderson, Avery, etc.[2-5]. Some preliminary definitions and theorems on time scales can be found in books [3,4] which are excellent references for calculus of time scales.

There have been extensive studies on a boundary value problem (BVP) with sign-changing nonlinearity on time scales by using the fixed point theorem on cones. See [6,7] and references therein. In [8], Feng, Pang and Ge discussed the existence of triple symmetric positive solutions by applying the fixed point theorem of functional type in a cone.

In [9], Ji, Bai and Ge studied the following singular multipoint boundary value problem:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M5">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6">View MathML</a>. By using fixed point index theory [10] and the Legget-Williams fixed point theorem [11], sufficient conditions for the existence of countably many positive solutions are established.

Sun, Wang and Fan [12] studied the nonlocal boundary value problem with p-Laplacian of the form

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M13">View MathML</a>. By using the four functionals fixed point theorem and five functionals fixed point theorem, they obtained the existence criteria of at least one positive solution and three positive solutions.

Inspired by the mentioned works, in this paper we consider the following m-point boundary value problem with an increasing homeomorphism and homomorphism:

(1.1)

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M16">View MathML</a> is a time scale, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M17">View MathML</a> is an increasing homeomorphism and homomorphism with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M18">View MathML</a>. A projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M19">View MathML</a> is called an increasing homeomorphism and homomorphism if the following conditions are satisfied:

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M21">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M22">View MathML</a>;

(ii) ϕ is a continuous bijection and its inverse mapping is also continuous;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M23">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M25">View MathML</a>.

We assume that the following conditions are satisfied:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M31">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M32">View MathML</a> is symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M34">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M35">View MathML</a>);

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M36">View MathML</a> symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M38">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M39">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M40">View MathML</a> on any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>.

By using four functionals fixed point theorem [5], we establish the existence of at least one symmetric positive solution for BVP (1.1)-(1.2). In particular, the nonlinear term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M42">View MathML</a> is allowed to change sign. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary lemmas. We give and prove our main result in Section 3. Section 4 contains an illustrative example. To the best of our knowledge, symmetric positive solutions for multipoint BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales by using four functionals fixed point theorem [5] have not been considered till now. In this paper, we intend to fill in such gaps in the literature.

In this paper, a symmetric positive solution x of (1.1) and (1.2) means a solution of (1.1) and (1.2) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45">View MathML</a>.

2 Preliminaries

To prove the main result in this paper, we will employ several lemmas. These lemmas are based on the BVP

(2.1)

(2.2)

Lemma 2.1If condition (H1) holds, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M48">View MathML</a>, boundary value problem (2.1) and (2.2) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M50">View MathML</a>

(2.3)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M51">View MathML</a>

(2.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M52">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M53">View MathML</a>

(2.5)

Proof Suppose x is a solution of BVP (2.1), (2.2). Integrating (2.1) from 0 to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M54">View MathML</a>

(2.6)

Integrating (2.6) from 0 to t, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M55">View MathML</a>

or integrating the same equation from t to 1, we achieve

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M56">View MathML</a>

Using boundary condition (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M57">View MathML</a>

(2.7)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M58">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M52">View MathML</a> satisfies (2.5).

On the other hand, it is easy to verify that if x is the solution of (2.3) or (2.4), then x is a solution of BVP (2.1), (2.2). The proof is accomplished. □

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M60">View MathML</a>is nonnegative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62">View MathML</a>on any subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>, then there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M64">View MathML</a>satisfying (2.5). Moreover, there is a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M65">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M66">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M67">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M68">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M69">View MathML</a> is continuous and strictly increasing. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M70">View MathML</a>

Therefore there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M71">View MathML</a> satisfying (2.5). Furthermore, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M72">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M73">View MathML</a> is continuous and strictly increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M76">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M77">View MathML</a>

implies that there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M65">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M79">View MathML</a>. Lemma is proved. □

Remark 2.1 By Lemmas 2.1 and 2.2, the unique solution of BVP (2.1), (2.2) can be rewritten in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M80">View MathML</a>

(2.9)

Lemma 2.3Let (H1) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M81">View MathML</a>is nonnegative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62">View MathML</a>on any subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>, then the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a>of BVP (2.1)-(2.2) has the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a>is concave on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M88">View MathML</a>,

(iii) there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M89">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M90">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M91">View MathML</a>.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a> is a solution of BVP (2.1)-(2.2), then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M94">View MathML</a> is nonincreasing so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M95">View MathML</a> is nonincreasing. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a> is concave.

(ii) We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M98">View MathML</a>. Furthermore, we get

If we continue like this, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M100">View MathML</a>

Using (H1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M101">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M102">View MathML</a>. Similarly,

If we continue in this way, we attain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M104">View MathML</a>

Using (H1), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M106">View MathML</a>. Therefore, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M108">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M110">View MathML</a> imply that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M111">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M90">View MathML</a>.

If there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M114">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M115">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M116">View MathML</a>

which is a contradiction.

(iv) From Lemmas 2.1 and 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M117">View MathML</a>. Hence we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M118">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M91">View MathML</a>.

The lemma is proved. □

Lemma 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M60">View MathML</a>is symmetric nonnegative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M62">View MathML</a>on any subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>, then the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a>of (2.1), (2.2) is concave and symmetric with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M125">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>.

Proof Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a> is concave and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M125">View MathML</a> from Lemma 2.3. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a> is symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>. For the symmetry of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M131">View MathML</a>, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M132">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M133">View MathML</a>. Therefore, from (2.9) and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M134">View MathML</a>, by the transformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M136">View MathML</a>

Again, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M137">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M138">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M49">View MathML</a> is symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>. The proof is accomplished. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M141">View MathML</a>. Then E is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M142">View MathML</a>

We define two cones by

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M144">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M145">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M146">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M147">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M148">View MathML</a>. Obviously, x is a solution of BVP (2.1)-(2.2) if and only if x is a fixed point of the operator F.

Lemma 2.5If (H1) holds, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M149">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M151">View MathML</a>

(2.10)

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150">View MathML</a>, one arrives at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M153">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M154">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M155">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M156">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M157">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M158">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M159">View MathML</a>

The proof is finalized. □

From Lemma 2.5, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M160">View MathML</a>

Lemma 2.6Suppose that (H1)-(H3) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M161">View MathML</a>is completely continuous.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M150">View MathML</a>. According to the definition of T and Lemma 2.3, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M163">View MathML</a>, which implies the concavity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M164">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>. On the other hand, from the definition of f and h, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M166">View MathML</a> holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M167">View MathML</a>, i.e., Tx is symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M33">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M169">View MathML</a>. By applying the Arzela-Ascoli theorem on time scales, we can obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M170">View MathML</a> is relatively compact. In view of the Lebesgue convergence theorem on time scales, it is obvious that T is continuous. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M171">View MathML</a> is a completely continuous operator. The proof is completed. □

3 Existence of one symmetric positive solution

Let α and Ψ be nonnegative continuous concave functionals on P, and let β and θ be nonnegative continuous convex functionals on P, then for positive numbers r, j, n and R, we define the sets:

(3.1)

Theorem 3.1[5]

IfPis a cone in a real Banach spaceE, αand Ψ are nonnegative continuous concave functionals onP, βandθare nonnegative continuous convex functionals onPand there exist positive numbersr, j, nandRsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M173">View MathML</a>

is a completely continuous operator, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a>is a bounded set. If

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M175">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M176">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M179">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M176">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M181">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M183">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M186">View MathML</a>;

(v) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M183">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M188">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185">View MathML</a>.

ThenAhas a fixed pointxin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a>.

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M191">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M192">View MathML</a>. For the convenience, we take the notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M193">View MathML</a>

and define the maps

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M194">View MathML</a>

(3.2)

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M197">View MathML</a> be defined by (3.1).

Theorem 3.2Assume (H1)-(H3) hold. If there exist constantsr, j, n, Rwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M199">View MathML</a>and suppose thatfsatisfies the following conditions:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M200">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M201">View MathML</a>;

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M202">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M203">View MathML</a>;

(C3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M204">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M205">View MathML</a>.

Then BVP (1.1)-(1.2) has at least one symmetric positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M206">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M207">View MathML</a>

Proof Boundary value problem (1.1)-(1.2) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M208">View MathML</a> if and only if x solves the operator equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M209">View MathML</a>. Thus we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T.

We first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M211">View MathML</a> is completely continuous. For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177">View MathML</a> with Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M213">View MathML</a>

which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a> is a bounded set. According to Lemma 2.6, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M211">View MathML</a> is completely continuous.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M216">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M217">View MathML</a>. By direct calculation,

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M219">View MathML</a>, which means that (i) in Theorem 3.1 is satisfied.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M222">View MathML</a>, we have from concavity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M223">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M224">View MathML</a>. Hence (ii) in Theorem 3.1 is fulfilled.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M181">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M178">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M227">View MathML</a>

and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M188">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M230">View MathML</a>

Thus (iii) and (v) in Theorem 3.1 hold true. We finally prove that (iv) in Theorem 3.1 holds.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M177">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M233">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M234">View MathML</a>

Thus, all conditions of Theorem 3.1 are satisfied. T has a fixed point x in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M174">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45">View MathML</a>. By condition (C3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M45">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M240">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M241">View MathML</a>. This means that x is a fixed point of the operator F. Therefore, BVP (1.1)-(1.2) has at least one symmetric positive solution. The proof is completed. □

4 An example

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M242">View MathML</a>. If we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M247">View MathML</a> in boundary value problem (1.1)-(1.2), then we have the following BVP on time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M16">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M249">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M250">View MathML</a>,

(4.2)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M253">View MathML</a>. By simple calculation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M254">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M258">View MathML</a>. It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M260">View MathML</a>.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M261">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M262">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M263">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M264">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M266">View MathML</a>.

So, all conditions of Theorem 3.2 hold. Thus, by Theorem 3.2, BVP (4.1) has at least one symmetric positive solution x such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/52/mathml/M267">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

References

  1. Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math.. 18, 18–56 (1990). Publisher Full Text OpenURL

  2. Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math.. 35, 3–22 (1999). Publisher Full Text OpenURL

  3. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston (2001)

  4. Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003)

  5. Avery, R, Henderson, J, O’Regan, D: Four functionals fixed point theorem. Math. Comput. Model.. 48, 1081–1089 (2008). Publisher Full Text OpenURL

  6. Hamal, NA, Yoruk, F: Symmetric positive solutions of fourth order integral BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales. Comput. Math. Appl.. 59, 3603–3611 (2010). Publisher Full Text OpenURL

  7. Xu, F, Meng, Z: The existence of positive solutions for third-order p-Laplacian m-point boundary value problems with sign changing nonlinearity on time scales. Adv. Differ. Equ.. 2009, Article ID 169321 (2009)

  8. Feng, H, Pang, H, Ge, W: Multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one-dimensional p-Laplacian. Nonlinear Anal.. 69, 3050–3059 (2008). Publisher Full Text OpenURL

  9. Ji, D, Bai, Z, Ge, W: The existence of countably many positive solutions for singular multipoint boundary value problems. Nonlinear Anal.. 72, 955–964 (2010). Publisher Full Text OpenURL

  10. Deimling, K: Nonlinear Functional Analysis, Springer, New York (1985)

  11. Leggett, RW, Williams, LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J.. 28, 673–688 (1979). Publisher Full Text OpenURL

  12. Sun, TT, Wang, LL, Fan, YH: Existence of positive solutions to a nonlocal boundary value problem with p-Laplacian on time scales. Adv. Differ. Equ.. 2010, Article ID 809497 (2010)