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Multiple positive solutions for a fourth-order integral boundary value problem on time scales

Yongkun Li* and Yanshou Dong

Author Affiliations

Department of Mathematics, Yunnan University Kunming, Yunnan 650091 People's Republic of China

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Boundary Value Problems 2011, 2011:59  doi:10.1186/1687-2770-2011-59


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/59


Received:9 November 2011
Accepted:29 December 2011
Published:29 December 2011

© 2011 Li and Dong; 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 article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

Keywords:
positive solutions; fixed points; integral boundary conditions; time scales

1 Introduction

Boundary value problem (BVP) for ordinary differential equations arise in different areas of applied mathematics and physics and so on, the existence and multiplicity of positive solutions for such problems have become an important area of investigation in recent years, lots of significant results have been established by using upper and lower solution arguments, fixed point indexes, fixed point theorems and so on (see [1-8] and the references therein). Especially, the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors (see [9-18] and the references therein).

However, the corresponding results for BVP with integral boundary conditions on time scales are still very few [19-21]. In this article, we discuss the multiple positive solutions for the following fourth-order system of integral BVP with a parameter on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M1">View MathML</a>

(1.1)

where ai, bi, ci, di ≥ 0, and ρi = aiciσ(T) + aidi + bici > 0(i = 1, 2), 0 < λ, μ < +∞, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M2">View MathML</a>, ℝ+ = [0, +∞), Ai and Bi are nonnegative and rd-continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M3">View MathML</a>.

The main purpose of this article is to establish some sufficient conditions for the existence of at least two positive solutions for system (1.1) by using the fixed point theorem of cone expansion and compression type. This article is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using the fixed point theorem of cone expansion and compression type, we establish sufficient conditions for the existence of at least two positive solutions for system (1.1). An illustrative example is given in Section 4.

2 Preliminaries

In this section, we will provide several foundational definitions and results from the calculus on time scales and give some lemmas which are used in the proof of our main results.

A time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4">View MathML</a> is a nonempty closed subset of the real numbers ℝ.

Definition 2.1. [22]For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M5">View MathML</a>, we define the forward jump operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M6">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M7">View MathML</a>, while the backward jump operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M8">View MathML</a>by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M9">View MathML</a>.

In this definition, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M11">View MathML</a>, where ∅, denotes the empty set. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M12">View MathML</a> and σ(t) = t, then t is called right-dense, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M13">View MathML</a> and ρ(t) = t, then t is called left-dense. We also need, below, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M14">View MathML</a>, which is derived from the time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M15">View MathML</a> as follows: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4">View MathML</a> has a left-scattered maximum m, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M16">View MathML</a>. Otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M17">View MathML</a>.

Definition 2.2. [22]Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M18">View MathML</a>is a function and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M19">View MathML</a>. Then x is called differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M20">View MathML</a>if there exists a θ ∈ ℝ such that for any given ε > 0, there is an open neighborhood U of t such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M21">View MathML</a>

In this case, xΔ(t) is called the delta derivative of x at t. The second derivative of x(t) is defined by xΔΔ(t) = (xΔ)Δ(t).

In a similar way, we can obtain the fourth-order derivative of x(t) is defined by x(4Δ)(t) = (((xΔ)Δ)Δ)Δ(t).

Definition 2.3. [22]A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M22">View MathML</a>is called rd-continuous provided it is continuous at right-dense points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4">View MathML</a>and its left-sided limits exist at left-dense points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M4">View MathML</a>. The set of rd-continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M23">View MathML</a>will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M24">View MathML</a>.

Definition 2.4. [22]A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M25">View MathML</a>is called a delta-antiderivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M26">View MathML</a>provide FΔ(t) = f(t) holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M27">View MathML</a>. In this case we define the integral of f by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M28">View MathML</a>

For convenience, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M30">View MathML</a> and for i = 1, 2, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M31">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M32">View MathML</a>

To establish the existence of multiple positive solutions of system (1.1), let us list the following assumptions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M33">View MathML</a>

In order to overcome the difficulty due to the dependence of f, g on derivatives, we first consider the following second-order nonlinear system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M34">View MathML</a>

(2.1)

where A0 is the identity operator, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M35">View MathML</a>

(2.2)

For the proof of our main results, we will make use of the following lemmas.

Lemma 2.1. The fourth-order system (1.1) has a solution (x, y) if and only if the nonlinear system (2.1) has a solution (u, v).

Proof. If (x, y) is a solution of the fourth-order system (1.1), let u(t) = xΔΔ(t), v(t) = yΔΔ(t), then it follows from the boundary conditions of system (1.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M36">View MathML</a>

Thus (u, v) = (xΔΔ(t), yΔΔ(t)) is a solution of the nonlinear system (2.1).

Conversely, if (u, v) is a solution of the nonlinear system (2.1), let x(t) = A2u(t), y(t) = A2v(t), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M37">View MathML</a>

which imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M38">View MathML</a>

Consequently, (x, y) = (A2u(t), A2v(t)) is a solution of the fourth-order system (1.1). This completes the proof.

Lemma 2.2. Assume that D11D21 ≠ 1 holds. Then for any h1 C(I', ℝ+), the following BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M39">View MathML</a>

(2.3)

has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M40">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M41">View MathML</a>

Proof. First suppose that u is a solution of system (2.3). It is easy to see by integration of BVP(2.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M42">View MathML</a>

(2.4)

Integrating again, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M43">View MathML</a>

(2.5)

Let t = σ(T) in (2.4) and (2.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M44">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M45">View MathML</a>

(2.7)

Substituting (2.6) and (2.7) into the second boundary value condition of system (2.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M46">View MathML</a>

(2.8)

From (2.8) and the first boundary value condition of system (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M47">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M48">View MathML</a>

(2.10)

Substituting (2.9) and (2.10) into (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M49">View MathML</a>

(2.11)

By (2.11), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M50">View MathML</a>

(2.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M51">View MathML</a>

(2.13)

By (2.12) and (2.13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M52">View MathML</a>

(2.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M53">View MathML</a>

(2.15)

Substituting (2.14) and (2.15) into (2.11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M54">View MathML</a>

(2.16)

Conversely, suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M55">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M56">View MathML</a>

(2.17)

Direct differentiation of (2.17) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M57">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M58">View MathML</a>

and it is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M59">View MathML</a>

This completes the proof.

Lemma 2.3. Assume that D12D22 ≠ 1 holds. Then for any h2 C(I', ℝ+), the following BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M60">View MathML</a>

has a solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M61">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M62">View MathML</a>

Proof. The proof is similar to that of Lemma 2.2 and will omit it here.

Lemma 2.4. Suppose that (H1) is satisfied, for all t, s I and i = 1, 2, we have

(i) Gi(t, s) > 0, Hi(t, s) > 0,

(ii) LimiGi(σ(s), s) ≤ Hi(t, s) ≤ MiGi(σ(s), s),

(iii) mGi(σ(s), s) ≤ Hi(t, s) ≤ MGi(σ(s), s),

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M63">View MathML</a>

Proof. It is easy to verify that Gi(t, s) > 0, Hi(t, s) > 0 and Gi(t, s) ≤ Gi(σ(s), s), for all t, s I. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M64">View MathML</a>

Thus Gi(t, s)/Gi(σ(s), s) ≥ Li and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M65">View MathML</a>

On the one hand, from the definition of Li and mi, for all t, s I, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M66">View MathML</a>

and on the other hand, we obtain easily that from the definition of Mi, for all t, s I,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M67">View MathML</a>

Finally, it is easy to verify that mGi(σ(s), s) ≤ Hi(t, s) ≤ MGi(σ(s), s). This completes the proof.

Lemma 2.5. [23]Let E be a Banach space and P be a cone in E. Assume that Ω1 and Ω2 are bounded open subsets of E, such that 0 ∈ Ω1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M68">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M69">View MathML</a>be a completely continuous operator such that either

(i) ||Tu|| ≤ ||u||, ∀u P ∩ ∂Ω1 and ||Tu|| ≥ ||u||, ∀u P ∩ ∂Ω2, or

(ii) ||Tu|| ≥ ||u||, ∀u P ∩ ∂Ω1 and ||Tu|| ≤ ||u||, ∀u P ∩ ∂Ω2

holds. Then T has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M70">View MathML</a>.

To obtain the existence of positive solutions for system (2.1), we construct a cone P in the Banach space Q = C(I, ℝ+) × C(I, ℝ+) equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M71">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M72">View MathML</a>

It is easy to see that P is a cone in Q.

Define two operators Tλ, Tμ : P C(I, ℝ+) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M73">View MathML</a>

Then we can define an operator T : P C(I, ℝ+) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M74">View MathML</a>

Lemma 2.6. Let (H1) hold. Then T : P P is completely continuous.

Proof. Firstly, we prove that T : P P. In fact, for all (u, v) ∈ P and t I, by Lemma 2.4(i) and (H1), it is obvious that Tλ(u, v)(t) > 0, Tμ(u, v)(t) > 0. In addition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M75">View MathML</a>

(2.18)

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M76">View MathML</a>. And we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M77">View MathML</a>

In a similar way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M78">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M79">View MathML</a>

This shows that T : P P.

Secondly, we prove that T is continuous and compact, respectively. Let {(uk, vk)} ∈ P be any sequence of functions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M80">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M81">View MathML</a>

from the continuity of f, we know that ||Tλ(uk, vk) - Tλ(u, v)|| → 0 as k → ∞. Hence Tλ is continuous.

Tλ is compact provided that it maps bounded sets into relatively compact sets. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M82">View MathML</a>, and let Ω be any bounded subset of P, then there exists r > 0 such that ||(u, v)|| ≤ r for all (u, v) ∈ Ω. Obviously, from (2.16), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M83">View MathML</a>

so, TλΩ is bounded for all (u, v) ∈ Ω. Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M84">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M85">View MathML</a>

Thus, for any (u, v) ∈ Ω and ∀ε > 0, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M86">View MathML</a>, then for t1, t2 I, |t1 - t2| < δ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M87">View MathML</a>

So, for all (u, v) ∈ Ω, TλΩ is equicontinuous. By Ascoli-Arzela theorem, we obtain that Tλ : P P is completely continuous. In a similar way, we can prove that Tμ : P P is completely continuous. Therefore, T : P P is completely continuous. This completes the proof.

3 Main results

In this section, we will give our main results on multiplicity of positive solutions of system (1.1). In the following, for convenience, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M88">View MathML</a>

where qi(t), qj(t) ∈ Crd(I', ℝ+) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M89">View MathML</a>

Theorem 3.1. Assume that (H1) holds. Assume further that

(H2) there exist a constant R > 0, and two functions pi(t) ∈ Crd(I, R+) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M90">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M91">View MathML</a>

and one of the folloeing conditions is satisfied

(E1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M93">View MathML</a>,

(E2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M95">View MathML</a>,

(E3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M96">View MathML</a>, μ ∈ (0, N4),

(E4) λ ∈ (0, M4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M97">View MathML</a>,

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M98">View MathML</a>

O1, O2 satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M99">View MathML</a>. Then system (1.1) has at least two positive solutions.

Proof. We only prove the case in which (H2) and (E1) hold, the other case can be proved similarly. Firstly, from (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M100">View MathML</a>

(3.1)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M101">View MathML</a>, and let Ω1 = {(u, v) ∈ Q; ||(u, v)|| < R1}. For any t I, (u, v) ∈ ∂Ω1 P, it follows from λ < M4, μ < N4 and (H2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M102">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M103">View MathML</a>

Consequently, for any (u, v) ∈ ∂Ω1 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M104">View MathML</a>

(3.2)

Second, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M105">View MathML</a>, we can choose ε1 > 0 such that λf0 > M3 + ε1, then there exists 0 < l1 < NR1 such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M106">View MathML</a> and t I,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M107">View MathML</a>

And since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M108">View MathML</a>

(3.3)

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M109">View MathML</a>

For all (u, v) ∈ Ω2 P, where Ω2 = {(u, v) ∈ Q; ||(u, v)|| < R2}, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M110">View MathML</a>

Thus, for all (u, v) ∈ Ω2 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M111">View MathML</a>

Consequently, for all (u, v) ∈ Ω2 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M112">View MathML</a>

(3.4)

Finally, from μ > N3/g, we can choose ε2 > 0 such that μg> N3 + ε2. then, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M113">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M114">View MathML</a> and t I,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M115">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M116">View MathML</a>

For all (u, v) ∈ Ω3 P, where Ω3 = {(u, v) ∈ Q; ||(u, v)|| < R3}, from (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M117">View MathML</a>

(3.5)

Thus, for all (u, v) ∈ Ω3 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M118">View MathML</a>

Consequently, for all (u, v) ∈ Ω3 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M119">View MathML</a>

(3.6)

From (3.2), (3.4), and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with R2 ≤ ||(u1, v1)|| ≤ R1. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x1, y1). In the same way, from (3.2), (3.6), and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with R1 ≤ ||(u2, v2)|| ≤ R3. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the proof.

Theorem 3.2. Assume that (H1) holds. Suppose further that

(H3) there exist a constant R0 > 0, and two functions wi(t) ∈ Crd(I, R+) (i = 1, 2) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M120">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M121">View MathML</a>

(3.7)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M122">View MathML</a>

(3.8)

Then system (1.1) has at least two positive solutions for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M123">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M124">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M125">View MathML</a>

Proof. We only prove the case in which (3.7) holds. The other case in which (3.8) holds can be proved similarly.

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M126">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M127">View MathML</a>. For any t I, (u, v) ∈ ∂Ω4 P, it follows from λ > M5 and (H3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M128">View MathML</a>

Consequently, for any (u, v) ∈ ∂Ω4 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M129">View MathML</a>

(3.9)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M131">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M133">View MathML</a>, we can choose ε3 > 0 such that M6 - ε3 > 0, N6 - ε3 > 0 and λf0 < M6 - ε3, μg0 < N6 - ε3. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M134">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M135">View MathML</a> and t I,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M136">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M138">View MathML</a>. Then, for any (u, v) ∈ Ω5 P, from(3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M139">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M140">View MathML</a>

Consequently, for any (u, v) ∈ ∂Ω5 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M141">View MathML</a>

(3.10)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M143">View MathML</a> we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M145">View MathML</a>, we can choose ε4 > 0 such that M6 - ε4 > 0, N6 - ε4 > 0 and λf< M6 - ε4, μg< N6 - ε4. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M146">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M147">View MathML</a> and t I,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M148">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M149">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M150">View MathML</a>. Then, for any (u, v) ∈ Ω6 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M151">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M152">View MathML</a>

Consequently, for any (u, v) ∈ ∂Ω6 P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M153">View MathML</a>

(3.11)

From (3.9), (3.10) and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M154">View MathML</a>. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x1, y1). In the same way, from (3.9), (3.11) and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M155">View MathML</a>. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the proof.

4 An example

Consider the following BVP with integral boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M156">View MathML</a>

(4.1)

where A1(t) = B1(t) = t, A2(t) = B2(t) = t/2 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M157">View MathML</a>

we choose O1 = 2, O2 = 4, R = 1, p1(t) = 2t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/59/mathml/M158">View MathML</a>, q1(t) = q3(t) = 1. It is easy to check that f0 = g= ∞, (H1), (H2) and (E1) are satisfied. Therefore, by Theorem 3.1, system (4.1) has at least two positive solutions for each λ ∈ (0, M4), μ ∈ (0, N4).

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

This study was supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

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