SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

Eigenvalues for iterative systems of nonlinear m-point boundary value problems on time scales

Ilkay Y Karaca1* and Fatma Tokmak12

Author Affiliations

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

2 Department of Mathematics, Gazi University, Teknikokullar, Ankara, 06500, Turkey

For all author emails, please log on.

Boundary Value Problems 2014, 2014:63  doi:10.1186/1687-2770-2014-63

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/63


Received:3 October 2013
Accepted:3 March 2014
Published:21 March 2014

© 2014 Karaca and Tokmak; licensee Springer.

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

Abstract

In this paper, we determine the eigenvalue intervals of the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M1">View MathML</a> for which there exist positive solutions of the iterative systems of m-point boundary value problems on time scales. The method involves an application of Guo-Krasnosel’skii fixed point theorem. We give an example to demonstrate our main results.

MSC: 34B18, 34N05.

Keywords:
Green’s function; iterative system; eigenvalue interval; time scales; boundary value problem; fixed point theorem; m-point; positive solution

1 Introduction

The study of dynamic equations on time scales goes back to Stefan Hilger [1]. Theoretically, this new theory has not only unify continuous and discrete equations, but it has also exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic equations on time scales has led to several important applications, for example, insect population models, biology, neural networks, heat transfer, and epidemic models; see [2-7].

There has been much interest shown in obtaining optimal eigenvalue intervals for the existence of positive solutions of the boundary value problems on time scales, often using Guo-Krasnosel’skii fixed point theorem. To mention a few papers along these lines, see [8-12]. On the other hand, there is not much work concerning the eigenvalues for iterative system of nonlinear boundary value problems on time scales; see [13,14].

In [15], Ma and Thompson are concerned with determining values λ, by using the Guo-Krasnosel’skii fixed point theorem for which there exist positive solutions of the m-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M2">View MathML</a>

In [13], Benchohra et al. studied the eigenvalues for iterative system of nonlinear boundary value problems on time scales,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M3">View MathML</a>

satisfying the boundary conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M4">View MathML</a>

The method involves application of Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space.

In [14], Prasad et al. studied the eigenvalues for iterative system of nonlinear boundary value problems on time scales,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M5">View MathML</a>

satisfying the m-point boundary conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M6">View MathML</a>

They used the Guo-Krasnosel’skii fixed point theorem.

Motivated by the above results, in this study, we are concerned with determining the eigenvalue intervals of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, for which there exist positive solutions for the iterative system of nonlinear m-point boundary value problems on time scales,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M9">View MathML</a>

(1.1)

satisfying the m-point boundary conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M10">View MathML</a>

(1.2)

where is a time scale, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M13">View MathML</a>.

Throughout this paper we assume that following conditions hold:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M14">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M15">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M17">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M18">View MathML</a>,

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M19">View MathML</a> is continuous, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>,

(C3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M22">View MathML</a> does not vanish identically on any closed subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>,

(C4) each of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, exists as positive real number.

In fact, our results are also new when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M28">View MathML</a> (the differential case) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M29">View MathML</a> (the discrete case). Therefore, the results can be considered as a contribution to this field.

This paper is organized as follows. In Section 2, we construct the Green’s function for the homogeneous problem corresponding to (1.1)-(1.2) and estimate bounds for the Green’s function. In Section 3, we determine the eigenvalue intervals for which there exist positive solutions of the boundary value problem (1.1)-(1.2) by using the Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space. Finally, in Section 4, we give an example to demonstrate our main results.

2 Preliminaries

We need the auxiliary lemmas that will be used to prove our main results.

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M30">View MathML</a>, which is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M31">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M32">View MathML</a>, then we consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M33">View MathML</a>

(2.1)

Denote by θ and φ, the solutions of the corresponding homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M34">View MathML</a>

(2.2)

under the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M35">View MathML</a>

(2.3)

Using the initial conditions (2.3), we can deduce from equation (2.2) for θ and φ the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M36">View MathML</a>

(2.4)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M37">View MathML</a>

(2.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M38">View MathML</a>

(2.6)

Lemma 2.1Let (C1) hold. Assume that

(C5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M39">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40">View MathML</a>is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M41">View MathML</a>

(2.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M42">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M43">View MathML</a>

(2.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M44">View MathML</a>

(2.10)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45">View MathML</a>is a solution of the boundary value problem (2.1).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45">View MathML</a> satisfy the integral equation (2.7), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M47">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M48">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M49">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M50">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M51">View MathML</a>

(2.11)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M52">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M53">View MathML</a>

(2.12)

From (2.11) and (2.12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M54">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M56">View MathML</a> satisfy (2.9) and (2.10), respectively. □

Lemma 2.2Let (C1) hold. Assume

(C6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M59">View MathML</a>.

Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M61">View MathML</a>, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45">View MathML</a>of the problem (2.1) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M63">View MathML</a>

Proof It is an immediate subsequence of the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M64">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M67">View MathML</a>. □

Lemma 2.3Let (C1) and (C6) hold. Assume

(C7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M68">View MathML</a>.

Then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M40">View MathML</a>of the problem (2.1) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M70">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71">View MathML</a>.

Proof Assume that the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M72">View MathML</a> holds. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M73">View MathML</a> is nonincreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23">View MathML</a>, one can verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M75">View MathML</a>

From the boundary conditions of the problem (2.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M76">View MathML</a>

The last inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M77">View MathML</a>

Therefore, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M78">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M79">View MathML</a>

According to Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M80">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M81">View MathML</a>. However, this contradicts to condition (C7). Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M70">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71">View MathML</a>. □

Lemma 2.4Let (C1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M61">View MathML</a>hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M85">View MathML</a>be a constant. Then the unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M45">View MathML</a>of the problem (2.1) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M87">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M88">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M89">View MathML</a>

(2.13)

Proof We have from (2.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M90">View MathML</a>

(2.14)

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M91">View MathML</a>

(2.15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71">View MathML</a>. Applying (2.8), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M94">View MathML</a>

(2.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M95">View MathML</a>

Thus for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M97">View MathML</a>

So, the proof is completed. □

We note that an n-tuple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M98">View MathML</a> is a solution of the boundary value problem (1.1)-(1.2) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M100">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M101">View MathML</a>

To determine the eigenvalue intervals of the boundary value problem (1.1)-(1.2), we will use the following Guo-Krasnosel’skii fixed point theorem [16].

Theorem 2.1[16]

Letbe a Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M103">View MathML</a>be a cone in. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M105">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M106">View MathML</a>are open subsets ofwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M108">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M109">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M110">View MathML</a>

be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M112">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M114">View MathML</a>, or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M116">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M118">View MathML</a>.

ThenThas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M119">View MathML</a>.

3 Positive solutions in a cone

In this section, we establish criteria to determine the eigenvalue intervals for which the boundary value problem (1.1)-(1.2) has at least one positive solution in a cone. We construct a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M103">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M121">View MathML</a>

where γ is given in (2.13).

Now, we define an integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M122">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a>, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M124">View MathML</a>

(3.1)

Notice from (C1)-(C6) and Lemma 2.2 that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M126">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M71">View MathML</a>. Also, we have from (2.8), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M128">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M129">View MathML</a>

(3.2)

Next, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a>, we have from Lemma 2.4 and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M131">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M133">View MathML</a>. In addition, the operator T is completely continuous by an application of the Arzela-Ascoli theorem.

Now, we investigate suitable fixed points of T belonging to the cone . For convenience we introduce the following notations.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M135">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M136">View MathML</a>

Theorem 3.1Suppose conditions (C1)-(C7) are satisfied. Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M137">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M138">View MathML</a>

(3.3)

there exists ann-tuple<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M139">View MathML</a>satisfying (1.1)-(1.2) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>, on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M144">View MathML</a>, be as in (3.3). Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M145">View MathML</a> be chosen such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M146">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M147">View MathML</a>

We investigate fixed points of the completely continuous operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M148">View MathML</a> defined by (3.1). Now, from the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M151">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M153">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M155">View MathML</a>. We have from (2.14) and the choice of ϵ, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M156">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M157">View MathML</a>

It follows in a similar manner from (2.14), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M158">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M159">View MathML</a>

Continuing with this bootstrapping argument, we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M160">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M161">View MathML</a>

so that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M160">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M163">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M164">View MathML</a>. If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M165">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M166">View MathML</a>

(3.4)

Next, from the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M169">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M171">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M172">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M174">View MathML</a>. Then, we have from Lemma 2.4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M175">View MathML</a>

Consequently, from Lemma 2.4 and the choice of ϵ, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M156">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M177">View MathML</a>

It follows in a similar manner from Lemma 2.4 and the choice of ϵ, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M179">View MathML</a>

Again, using a bootstrapping argument, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M180">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M181">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M182">View MathML</a>. So if we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M183">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M184">View MathML</a>

(3.5)

Applying Theorem 2.1 to (3.4) and (3.5), we see that T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M185">View MathML</a>. Therefore, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M186">View MathML</a>, we obtain a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M187">View MathML</a> of (1.1)-(1.2) given iteratively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M188">View MathML</a>

The proof is completed. □

For our next result, we define the positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M190">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M191">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M192">View MathML</a>

Theorem 3.2Suppose conditions (C1)-(C7) are satisfied. Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M137">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M194">View MathML</a>

(3.6)

there exists ann-tuple<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M139">View MathML</a>satisfying (1.1)-(1.2) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>, on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M23">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M144">View MathML</a>, be as in (3.6). Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M145">View MathML</a> be chosen such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M202">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M203">View MathML</a>

Let T be the cone preserving, completely continuous operator that was defined by (3.1). From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M205">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M206">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M208">View MathML</a>

Also, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M149">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M24">View MathML</a>, and so there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M212">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M213">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M214">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M216">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M217">View MathML</a>

Continuing with this bootstrapping argument, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M218">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M219">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M182">View MathML</a>. If we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M221">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M222">View MathML</a>

(3.7)

Since each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167">View MathML</a> is assumed to be a positive real number, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M224">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, is unbounded at ∞.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M227">View MathML</a>

Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M229">View MathML</a> is a nondecreasing real-valued function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M230">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M231">View MathML</a>

Next, by definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M234">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M236">View MathML</a>

It follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M237">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M239">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M123">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M241">View MathML</a>. Then, using the bootstrapping argument, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M242">View MathML</a>

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M243">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M244">View MathML</a>. So, if we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M245">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M246">View MathML</a>

(3.8)

Applying Theorem 2.1 to (3.7) and (3.8), we see that T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M247">View MathML</a>, which in turn with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M186">View MathML</a>, we obtain an n-tuple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M187">View MathML</a> satisfying (1.1)-(1.2) for the chosen values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M8">View MathML</a>. The proof is completed. □

4 An example

Example 4.1 In BVP (1.1)-(1.2), suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M260">View MathML</a>i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M261">View MathML</a>

(4.1)

satisfying the following boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M262">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M263">View MathML</a>

It is easy to see that (C1)-(C7) are satisfied. By simple calculation, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M270">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M271">View MathML</a>

We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M272">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M273">View MathML</a>

Applying Theorem 3.1, we get the optimal eigenvalue interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/63/mathml/M275">View MathML</a>, for which the boundary value problem (4.1)-(4.2) has a positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

References

  1. Hilger, S: Ein maßkettenkalkül mit anwendug auf zentrumsmanningfaltigkeite. PhD thesis, Universität Würzburg (1988)

  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. Anderson, DR, Karaca, IY: Higher-order three-point boundary value problem on time scales. Comput. Math. Appl.. 56, 2429–2443 (2008). Publisher Full Text OpenURL

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

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

  6. Bohner, M, Luo, H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Adv. Differ. Equ.. 2006, Article ID 54989 (2006)

  7. Tokmak, F, Karaca, IY: Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales. Bound. Value Probl.. 2013, Article ID 52 (2013)

  8. Agarwal, RP, Bohner, M, Wong, P: Strum-Liouville eigenvalue problems on time scale. Appl. Math. Comput.. 99, 153–166 (1999). Publisher Full Text OpenURL

  9. Anderson, DR: Eigenvalue intervals for even order Strum-Liouville dynamic equations. Commun. Appl. Nonlinear Anal.. 12, 1–13 (2005)

  10. Benchohra, M, Henderson, J, Ntouyas, SK: Eigenvalue problems for systems of nonlinear boundary value problems on time scales. Adv. Differ. Equ.. 2007, Article ID 31640 (2007)

  11. Karaca, IY: Multiple positive solutions for dynamic m-point boundary value problems. Dyn. Syst. Appl.. 17, 25–42 (2008)

  12. Karaca, IY: Existence and nonexistence of positive solutions to a right-focal boundary value problem on time scales. Adv. Differ. Equ.. 2006, Article ID 43039 (2006)

  13. Benchohra, M, Berhoun, F, Hamani, S, Henderson, J, Ntouyas, SK, Ouahab, A, Purnaras, IK: Eigenvalues for iterative systems of nonlinear boundary value problems on time scales. Nonlinear Dyn. Syst. Theory. 9, 11–22 (2009)

  14. Prasad, KR, Sreedhar, N, Narasimhulu, Y: Eigenvalue intervals for iterative systems of nonlinear m-point boundary value problems on time scales. Differ. Equ. Dyn. Syst. (2013). Publisher Full Text OpenURL

  15. Ma, R, Thompson, B: Positive solutions for nonlinear m-point eigenvalue problems. J. Math. Anal. Appl.. 297, 24–37 (2004). Publisher Full Text OpenURL

  16. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, Orlando (1988)