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Multiple positive solutions for first-order impulsive integral boundary value problems on time scales

Yongkun Li* and Jiangye Shu

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

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Boundary Value Problems 2011, 2011:12  doi:10.1186/1687-2770-2011-12


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/12


Received:10 March 2011
Accepted:15 August 2011
Published:15 August 2011

© 2011 Li and Shu; licensee Springer.

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

Abstract

In this paper, we first present a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, some criteria for the existence of at least one, two, and three positive solutions are established for the problem under consideration, respectively. Finally, examples are presented to illustrate the main results.

MSC: 34B10; 34B37; 34N05.

Keywords:
integral boundary value problem; fixed point; multiple solutions; time scale

1 Introduction

In fact, continuous and discrete systems are very important in implementing and applications. It is well known that the theory of time scales has received a lot of attention, which was introduced by Stefan Hilger in order to unify continuous and discrete analyses. Therefore, it is meaningful to study dynamic systems on time scales, which can unify differential and difference systems.

In recent years, a great deal of work has been done in the study of the existence of solutions for boundary value problems on time scales. For the background and results, we refer the reader to some recent contributions [1-5] and references therein. At the same time, boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [6-12], since such equations may exhibit several real-world phenomena in physics, biology, engineering, etc. see [13-15] and the references therein.

In paper [16], Sun studied the first-order boundary value problem on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M1">View MathML</a>

(1.1)

where 0 < β < 1. By means of the twin fixed point theorem due to Avery and Henderson, some existence criteria for at least two positive solutions were established.

Tian and Ge [17] studied the first-order three-point boundary value problem on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M2">View MathML</a>

(1.2)

Using several fixed point theorems, the existence of at least one positive solution and multiple positive solutions is obtained.

However, except BVP of differential and difference equations, that is, for particular time scales (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M3">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M4">View MathML</a>), there are few papers dealing with multi-point boundary value problems more than three-point for first-order systems on time scales. In addition, problems with integral boundary conditions arise naturally in thermal conduction problems [18], semiconductor problems [19], hydrodynamic problems [20]. In continuous case, since integral boundary value problems include two-point, three-point,..., n-point boundary value problems, such boundary value problems for continuous systems have received more and more attention and many results have worked out during the past ten years, see Refs. [21-27] for more details. To the best of authors' knowledge, up to the present, there is no paper concerning the boundary value problem with integral boundary conditions on time scales. This paper is to fill the gap in the literature.

In this paper, we are concerned with the following first-order nonlinear impulsive integral boundary value problem on time scales:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M5">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a> is a time scale which is a nonempty closed subset of ℝ with the topology and ordering inherited from ℝ, 0, and T are points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>, an interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M7">View MathML</a> which has finite right-scattered points, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M9">View MathML</a> and p is regressive, ℝ+), Ii(1 ≤ i m) ∈ C([0, +∞), [0, +∞)), g is a nonnegative integrable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M11">View MathML</a>, ep(0,σ(T)) is the exponential function on time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>, which will be introduced in the next section, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M12">View MathML</a>, 0 < t1 < · · · < tm < T, and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M14">View MathML</a> represent the right and left limits of x(t) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M15">View MathML</a>.

Remark 1.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M16">View MathML</a> denote the set of right-scattered points in interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M17">View MathML</a>, 0 ≤ θ1 < · · · < θq T, σ(θ0) = 0, θq+1 = T. By some basic concepts and time scale calculus formulae in the book by Bohner and Peterson [28], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M18">View MathML</a>

(1.4)

The main purpose of this paper is to establish some sufficient conditions for the existence of at least one, two, or three positive solutions for BVP (1.3) using Guo-Krasnoselskii and Legget-Williams fixed point theorem, respectively.

For convenience, we introduce the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M19">View MathML</a>

where i = 1, 2,..., m.

This paper is organized as follows. In Section 2, some basic definitions and lemmas on time scales are introduced without proofs. In Section 3, some useful lemmas are established. In particular, Green's function for BVP (1.3) is established. We prove the main results in Sections 4-6.

2 Preliminaries

In this section, we shall first recall some basic definitions, lemmas that are used in what follows. For the details of the calculus on time scales, we refer to books by Bohner and Peterson [28,29].

Definition 2.1. [28]A time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>is an arbitrary nonempty closed subset of the real set with the topology and ordering inherited from ℝ. The forward and backward jump operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M20">View MathML</a>and the graininess <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M21">View MathML</a>are defined, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M22">View MathML</a>

In this definition, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M23">View MathML</a> (i.e., σ(t) = t if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>has a maximum t) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M24">View MathML</a> (i.e., ρ(t) = t if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>has a minimum t). The point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M25">View MathML</a>is called left-dense, left-scattered, right-dense, or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, or σ(t) > t, respectively. Points that are right-dense and left-dense at the same time are called dense. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>has a left-scattered maximum m1, defined <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M26">View MathML</a>; otherwise, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M27">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>has a right-scattered minimum m2, defined <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M28">View MathML</a>, otherwise, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M27">View MathML</a>.

Definition 2.2. [28]A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29">View MathML</a>is rd continuous provided it is continuous at each right-dense point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>and has a left-sided limit at each left-dense point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>. The set of rd-continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29">View MathML</a>will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M30">View MathML</a>.

Definition 2.3. [28]If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29">View MathML</a>is a function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31">View MathML</a>, then the delta derivative of f at the point t is defined to be the number fΔ(t) (provided it exists) with the property that for each ε > 0 there is a neighborhood U of t such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M32">View MathML</a>

Definition 2.4. [28]For a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29">View MathML</a> (the range of f may be actually replaced by Banach space), the (delta) derivative is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M33">View MathML</a>

if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M34">View MathML</a>

provided this limit exists.

Definition 2.5. [28]If FΔ(t) = f(t), then we define the delta integral by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M35">View MathML</a>

Definition 2.6. [28]A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M36">View MathML</a>is said to be regressive provided 1 + μ(t)p(t) ≠ 0 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31">View MathML</a>, where μ(t) = σ(t) - t is the graininess function. The set of all regressive rd-continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M29">View MathML</a>is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M37">View MathML</a>, while the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M38">View MathML</a>is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M39">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M25">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M40">View MathML</a>. The exponential function is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M41">View MathML</a>

where ξh(z) is the so-called cylinder transformation.

Lemma 2.1. [28]Let p, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M42">View MathML</a>. Then

(1) e0(t, s) ≡ 1 and ep(t, t) ≡ 1;

(2) ep(σ(t), s) = (1 + μ(t)p(t))ep(t, s);

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M43">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M44">View MathML</a>;

(4) ep(t, s)ep(s, r) = ep(t, r),

(5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M45">View MathML</a>.

Lemma 2.2. [28]Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M46">View MathML</a>are delta differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M47">View MathML</a>

Lemma 2.3. [28]Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M49">View MathML</a>, and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M50">View MathML</a>is continuous at (t, t), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M31">View MathML</a>with t > a. Also, assume that fΔ(t, ·) is rd-continuous on [a, σ(t)]. Suppose that for each ε > 0 there exists a neighborhood U of t, independent of τ ∈ [a, σ(t)], such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M51">View MathML</a>

where fΔ denotes the derivative of f with respect to the first variable. Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M52">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M53">View MathML</a>.

3 Foundational lemmas

In this section, we first introduce some background definitions, fixed point theorems in Banach space, then present basic lemmas that are very crucial in the proof of the main results.

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M54">View MathML</a> is a piecewise continuous map with first-class discontinuous points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M55">View MathML</a> and at each discontinuous point it is continuous on the left} with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M56">View MathML</a>, then PC is a Banach Space.

Definition 3.1. A function x is said to be a positive solution of problem (1.3) if x PC satisfying problem (1.3) and x(t) > 0 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M57">View MathML</a>.

Definition 3.2. Let X be a real Banach space, the nonempty set K X is called a cone of X, if it satisfies the following conditions.

(1) x K and λ ≥ 0 implies λx K;

(2) x K and -x K implies x = 0.

Every cone K X induces an ordering in X, which is given by x y if and only if y - x K.

Definition 3.3. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Lemma 3.1. (Guo-Krasnoselskii [30]) Let X be a Banach space and K X be a cone in X. Assume that Ω1, Ω2 are bounded open subsets of X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M58">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M59">View MathML</a>is a completely continuous operator such that, either

(1) ||Φx|| ≤ ||x||, x K ∩ ∂Ω1, and ||Φx|| ≥ ||x||, x K ∩ ∂Ω2; or

(2) ||Φx|| ≥ ||x||, x K ∩ ∂Ω1, and ||Φx|| ≤ ||x||, x K ∩ ∂Ω2.

Then Φ has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M60">View MathML</a>.

Lemma 3.2. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M61">View MathML</a>, νi ∈ ℝ, then x is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M62">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M63">View MathML</a>

if and only if x is a solution of the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M64">View MathML</a>

(3.2)

Proof. Assume that x(t) is a solution of (3.2). By the first equation in (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M65">View MathML</a>

(3.3)

If t ∈ [0, t1], integrating (3.3) from 0 to t, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M66">View MathML</a>

while t t1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M67">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M68">View MathML</a>

Now, let t ∈ (t1, t2], integrating (3.3) from t1 to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M69">View MathML</a>

For t ∈ (tk, tk+1], repeating the above process, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M70">View MathML</a>

that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M71">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M72">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M74">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M75">View MathML</a>

(3.4)

This means that if x is a solution of (3.2) then x satisfies (3.1).

On the other hand, if x satisfies (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M76">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M77">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M78">View MathML</a>

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M79">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M80">View MathML</a>

Hence, we get from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M81">View MathML</a>

that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M82">View MathML</a>

Finally, we can obtain from (3.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M84">View MathML</a>

So the proof of this lemma is completed.

Lemma 3.3. Let G(t, s) be defined the same as that in Lemma 3.2, then the following properties hold.

(1) G(t, s) > 0 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M85">View MathML</a>;

(2) A G(t, s) ≤ B for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M85">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M86">View MathML</a>

Proof. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M87">View MathML</a>, then it is clear that (1) holds. Now we will show that (2) holds.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M88">View MathML</a>

Hence, the left-hand side of (2) holds. And it is easy to show that the right-hand side of (2) also holds. The proof is complete. ■

Define an operator Φ : PC PC by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M89">View MathML</a>

By Lemma 3.2, the fixed points of Φ are solutions of problem (1.3).

Lemma 3.4. The operator Φ : PC PC is completely continuous.

Proof. The first step we will show that Φ : PC PC is continuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M90">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M91">View MathML</a> in PC. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M92">View MathML</a>

Since f(t, x) and Ii(x)(1 ≤ i m) are continuous in x, we have |(Φxn)(t) - (Φx)(t)| → 0, which leads to ||Φxn - Φx||PC → 0, as n → ∞. That is, Φ : PC PC is continuous.

Next, we will show that Φ : PC PC is a compact operator by two steps.

Let U PC be a bounded set.

Firstly, we will show that {Φx : x U}is bounded. For any x U, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M93">View MathML</a>

In virtue of the continuity of f(t, x) and Ii(x)(1 ≤ i m), we can conclude that {Φx : x U} is bounded from above inequality.

Secondly, we will show that {Φx : x U} is the set of equicontinuous functions. For any x, y U, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M94">View MathML</a>

In virtue of the continuity of f(t, x) and Ii(x)(1 ≤ i m), the right-hand side tends to zero uniformly as |x - y| → 0. Consequently, {Φx : x U} is the set of equicontinuous functions.

By Arzela-Ascoli theorem on time scales [31], {Φx : x U} is a relatively compact set. So Φ maps a bounded set into a relatively compact set, and Φ is a compact operator.

From above three steps, it is easy to see that Φ : PC PC is completely continuous. The proof is complete. ■

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M95">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M96">View MathML</a>. It is not difficult to verify that K is a cone in PC.

Lemma 3.5. Φ maps K into K.

Proof. Obviously, Φ(K) ⊂ PC. ∀x K, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M97">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M98">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M99">View MathML</a>

Hence, Φ(K) ⊂ K. The proof is complete. ■

4 Existence of at least one positive solution

In this section, we will state and prove our main result about the existence of at least one positive solution of problem (1.3).

Theorem 4.1. Assume that one of the following conditions is satisfied:

(H1) max f0 = 0, min f= ∞, and Ii0 = 0, i = 1, 2,..., m; or

(H2) max f= 0, min f0 = ∞, and Ii= 0, i = 1, 2,..., m.

Then, problem (1.3) has at least one positive solution.

Proof. Firstly, we assume that (H1) holds. In this case, since max f0 = 0 and Ii0 = 0, i = 1, 2,..., m, for ε ≤ ((T) + Bm)-1, there exists a positive constant r1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M100">View MathML</a>

In view of min f= ∞, we have that for M ≥ ((T)δ)-1, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M101">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M102">View MathML</a>

Let Ωi = {x PC : ||x|| < ri}, i = 1, 2.

On the one hand, if x K ∩ ∂Ω1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M103">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M104">View MathML</a>

(4.1)

On the other hand, if x K ∩ ∂Ω2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M105">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M106">View MathML</a>

(4.2)

Therefore, by (4.1), (4.2), and Lemma 3.1, it follows that Φ has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M60">View MathML</a>.

Next, we assume that (H2) holds. In this case, since max f= 0 and Ii= 0, i = 1, 2,..., m, for ε' ≤ ((T) + Bm)-1, there exists a positive constant r3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M107">View MathML</a>

In view of min f= ∞, we have that for M' ≥ ((T)δ)-1, there exists a positive constant r4 < δr3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M108">View MathML</a>

Let Ωi = {x PC : ||x|| < ri}, i = 3, 4.

On the one hand, if x K ∩ ∂Ω3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M109">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M110">View MathML</a>

(4.3)

On the other hand, if x K ∩ ∂Ω4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M111">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M112">View MathML</a>

(4.4)

Hence, from (4.3) and (4.4) and Lemma 3.1, we conclude that Φ has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M113">View MathML</a>, that is, problem (1.3) has at least one positive solution. The proof is complete. ■

5 Existence of at least two positive solutions

In this section, we will state and prove our main results about the existence of at least two positive solutions to problem (1.3).

Theorem 5.1. Assume that the following conditions hold.

(H3) min f0 = +∞, min f= +∞.

(H4) There exists a positive constant R such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M114">View MathML</a>for all 0 < x R.

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M115">View MathML</a>, x ∈ (0, ∞), i = 1, 2,..., m.

Then, problem (1.3) has at least two positive solutions.

Proof. Let ΩR = {x PC : ||x|| < R}. From (H4) and (H5), for x K ∩ ∂ΩR, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M116">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M117">View MathML</a>

(5.1)

Since min f0 = +∞, for M ≥ ((T)δ)-1, there exists a positive constant R1 < δR such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M118">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M119">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M120">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M121">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M122">View MathML</a>

(5.2)

Similarly, since min f= +∞, for M' ≥ ((T)δ)-1, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M123">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M124">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M125">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M126">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M127">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M128">View MathML</a>

(5.3)

Equations 5.1 and 5.2 imply that Φ has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M129">View MathML</a>, which is a positive solution of problem (1.3). Besides, (5.1) and (5.3) imply that Φ has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M130">View MathML</a>, which is a positive solution of problem (1.3). Therefore, problem (1.3) has at least two positive solutions x1 and x2 satisfying 0 < R1 ≤ ||x1|| < R < ||x2|| ≤ R2. The proof is complete. ■

Theorem 5.2. Assume that the following conditions hold.

(H6) max f0 = 0, max f= 0, Ii0 = 0, Ii= 0, i = 1, 2,..., m.

(H7) There exists a positive constant r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M131">View MathML</a>for all 0 < x r.

Then problem (1.3) has at least two positive solutions.

Proof. Let Ωr = {x PC : ||x|| < r}. From (H7), for x K ∩ ∂Ωr, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M132">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M133">View MathML</a>

(5.4)

Since max f0 = 0 and Ii0 = 0, i = 1, 2,..., m, for ε ≤ ((T) + Bm)-1, there exists a positive constant r1 < δr such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M134">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M135">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M136">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M137">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M138">View MathML</a>

(5.5)

Similarly, since max f= 0 and Ii= 0, i = 1, 2,..., m, for ε' ≤ ((T) + Bm)-1, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M139">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M140">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M141">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M142">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M143">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M144">View MathML</a>

(5.6)

Equations 5.4 and 5.5 imply that Φ has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M145">View MathML</a>, which is a positive solution of problem (1.3). Besides, (5.4) and (5.6) imply that Φ has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M146">View MathML</a>, which is a positive solution of problem (1.3). Therefore, problem (1.3) has at least two positive solutions x1 and x2 satisfying 0 < r1 ≤ ||x1|| < r < ||x2|| ≤ r2. The proof is complete. ■

Similar to Theorems 5.1 and 5.2, one can easily obtain the following corollary:

Corollary 5.1. Assume that (H7) and the following conditions hold.

(H8) max f0 = 0, max f= 0, Ii0 = 0, i = 1, 2,..., m.

(H9) There exists a positive constant d such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M147">View MathML</a>for all x d, i = 1, 2,..., m.

Then, problem (1.3) has at least two positive solutions.

6 Existence of at least three positive solutions

In this section, we will state and prove our multiplicity result of positive solutions to problem (1.3) via Legget-Williams fixed point theorem. For readers' convenience, we first illustrate Legget-Williams fixed point theorem.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M148">View MathML</a> be a real Banach space with cone K. A map α : K → [0, +∞) is said to be a continuous concave functional on K if α is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M149">View MathML</a>

for all x, y K and t ∈ [0, 1]. Let a, b be two numbers such that 0 < a < b and α be a nonnegative continuous concave functional on K. We define the following convex sets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M150">View MathML</a>

Lemma 6.1. (Legget-Williams fixed point theorem [32]). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M151">View MathML</a>be completely continuous and α be a nonnegative continuous concave functional on K such that α(x) ≤ ||x|| for all x Kc. Suppose that there exist 0 < d < a < b c such that

(1) {x K(α, a, b) : α(x) > a} ≠ ∅, and α(Φ(x)) > a for all x K(α, a, b);

(2) ||Φx|| < d for all ||x|| ≤ d;

(3) α(Φ(x)) > a for all x K(α, a, c) with ||Φ(x)|| > b.

Then, Φ has at least three fixed points x1, x2, x3 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M152">View MathML</a>satisfying ||x1|| < d, a < α(x2), ||x3|| > d, and α(x3) < a.

Theorem 6.1. Assume that there exist numbers d, a, and c with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M153">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M154">View MathML</a>

(6.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M155">View MathML</a>

(6.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M156">View MathML</a>

(6.3)

Then, problem (1.3) has at least three positive solutions.

Proof. For x K, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M157">View MathML</a>

It is easy to verify that α is a nonnegative continuous concave functional on K with α(x) < ||x|| for all x K.

We first claim that if there exists a positive constant r such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M159">View MathML</a>, i = 1, 2,..., m, for x ∈ (0, r], then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M160">View MathML</a>.

Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M161">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M162">View MathML</a>

Thus, ||Φx|| < r, that is Φx Kr. Hence, we have shown that (6.1) or (6.2) hold, then Φ maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M163">View MathML</a> into Kd or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M152">View MathML</a> into Kc, respectively. So condition (2) of Lemma 6.1 holds.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M164">View MathML</a>. Next, we will show that {x K(α, a, b) : α(x) > a} ≠ ∅, and α(Φ(x)) > a for x K(α, a, b). In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M165">View MathML</a>, then the constant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M166">View MathML</a>.

Since (6.3) holds, for x K(α, a, b), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M167">View MathML</a>

So α(Φ(x)(t)) > a for all x K(α, a, b), then condition (1) of Lemma 6.1 holds.

Finally, suppose x K(α, a, c) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M168">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M169">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M170">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M171">View MathML</a>

To sum up, all the conditions of Theorem 6.1 are satisfied. Hence, Φ has at least three fixed points, that is, problem (1.3) has at least three positive solutions x1, x2, x3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M172">View MathML</a>

The proof is complete. ■

7 Examples

In this section, we give some examples to illustrate our main results.

Example 7.1. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173">View MathML</a>. We consider the following IBVP on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M174">View MathML</a>

(7.1)

where T = 3, p(t) = t, f(t, x(σ(t))) = (t + 1)(x(σ(t)))2, I(x) = x3, α = 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M175">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M176">View MathML</a>

From (1.4), system (7.1) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M177">View MathML</a>

By calculating, we get Γ = 0.3033 > 0, max f0 = 0, min f= ∞, and I0 = 0. Therefore, (H1) holds. From Theorem 4.1w, it follows that the IBVP (7.1) has at least one solution.

Example 7.2. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173">View MathML</a>. We consider the following IBVP on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M178">View MathML</a>

(7.2)

where p(t) = t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M180">View MathML</a>, α = 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M175">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M181">View MathML</a>

By calculating, we get Γ = 0.5732 > 0, max f= 0, min f0 = ∞, and I= 0. Therefore, by Theorem 4.1, it follows that the IBVP (7.2) has at least one solution.

Example 7.3. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M173">View MathML</a>. We consider the following IBVP on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M182">View MathML</a>

(7.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M183">View MathML</a>

Since p(t) = 1, T = 3, and σ(T) = 4, we know that ep(σ(T), 0) = 4e2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M184">View MathML</a>. Take R = 4976, then we can choose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M185">View MathML</a>

By calculating, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M186">View MathML</a>, I(x) ∈ C(ℝ0, ℝ0) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M187">View MathML</a>

Therefore, all the conditions of Theorem 5.1 are fulfilled. So system (7.3) has at least two positive solutions.

Example 7.4. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M188">View MathML</a>. We consider the following IBVP on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M6">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M189">View MathML</a>

(7.4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M190">View MathML</a>

Since p(t) = 1, T = 3, and σ(T) = 3, we know that ep(σ(T), 0) = 2e2. Then, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M191">View MathML</a>

Thus, if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/12/mathml/M193">View MathML</a>, and c is sufficiently large, then all the conditions of Theorem 6.1 are satisfied. So system (7.4) has at least three positive solutions.

8 Conclusion

In this paper, we first present a class of integral boundary value problems on time scales. Using the time scales calculus theory, the well-known Guo-Krasnoselskii fixed point theorem, and Legget-Williams fixed point theorem, we establish the existence of at least one, two, and three positive solutions for the problems. In addition, the methods in this paper may be applied to some other systems such as second-order integral boundary problems and higher-order integral boundary problems.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

In this paper, the authors first presented a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, by using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, they established some criteria for the existence of at least one, two, and three positive solutions to the problem under consideration, respectively. All authors typed, read and approved the final manuscript.

Acknowledgements

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

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