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Existence of positive solutions for nonlinear m-point boundary value problems on time scales

Junfang Zhao1*, Hairong Lian1 and Weigao Ge2

Author Affiliations

1 School of Science, China University of Geosciences, Beijing 100083, P.R. China

2 Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China

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Boundary Value Problems 2012, 2012:4  doi:10.1186/1687-2770-2012-4

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/4


Received:4 May 2011
Accepted:17 January 2012
Published:17 January 2012

© 2012 Zhao et al; 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 study the following m-point boundary value problem on time scales,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2">View MathML</a> is a time scale such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M3">View MathML</a>, ϕp(s) = |s|p-2s,p > 1,h Cld((0, T), (0, +∞)), and f C([0,+∞), (0,+∞)), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M4">View MathML</a>. By using several well-known fixed point theorems in a cone, the existence of at least one, two, or three positive solutions are obtained. Examples are also given in this article.

AMS Subject Classification: 34B10; 34B18; 39A10.

Keywords:
positive solutions; cone; multi-point; boundary value problem; time scale

1 Introduction

The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Further, the study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, epidemic models, etc. [2].

Multipoint boundary value problems of ordinary differential equations (BVPs for short) arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross section and composed of N parts of different densities can be set up as a multi-point boundary value problem [3]. Many problems in the theory of elastic stability can be handled by the method of multi-point problems [4]. Small size bridges are often designed with two supported points, which leads into a standard two-point boundary value condition and large size bridges are sometimes contrived with multi-point supports, which corresponds to a multi-point boundary value condition [5]. The study of multi-point BVPs for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [6]. Since then many authors have studied more general nonlinear multi-point BVPs, and the multi-point BVP on time scales can be seen as a generalization of that in ordinary differential equations.

Recently, the existence and multiplicity of positive solutions for nonlinear differential equations on time scales have been studied by some authors [7-11], and there has been some merging of existence of positive solutions to BVPs with p-Laplacian on time scales [12-19].

He [20] studied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M5">View MathML</a>

(1.1)

subject to one of the following boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M6">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M7">View MathML</a>. By using a double fixed-point theorem, the authors get the existence of at least two positive solutions to BVP (1.1) and (1.2).

Anderson [21] studied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M8">View MathML</a>

(1.3)

subject to one of the following boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M9">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M10">View MathML</a>

(1.5)

by using a functional-type cone expansion-compression fixed-point theorem, the author gets the existence of at least one positive solution to BVP (1.3), (1.4) and BVP (1.3), (1.5).

However, to the best of the authors' knowledge, up to now, there are few articles concerned with the existence of m-point boundary value problem with p-Laplacian on time scales. So, in this article, we try to fill this gap. Motivated by the article mentioned above, in this article, we consider the following m-point BVP with one-dimensional p-Laplacian,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M11">View MathML</a>

(1.6)

where ϕp(s) = |s|p-2s,p > 1,h Cld((0,T), (0, +∞)), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M12">View MathML</a>. δ, βi > 0, i = 1,..., m - 2.

We will assume throughout

(S1) h ∈ Cld ((0, T), [0, ∞)) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M13">View MathML</a>;

(S2) f C([0, ∞), (0, ∞)), f ≢ 0 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M14">View MathML</a>;

(S3) By ϕq we denote the inverse to ϕp, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M15">View MathML</a>;

(S4) By t ∈ [a, b] we mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M16">View MathML</a>, where 0 ≤ a b T.

2 Preliminaries

In this section, we will give some background materials on time scales.

Definition 2.1. [7,22] For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M18">View MathML</a>, define the forward jump operator σ and the backward jump operator ρ, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M19">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M20">View MathML</a>. If σ(t) > t, t is said to be right scattered, and if ρ(r) < r, r is said to be left scattered. If σ(t) = t, t is said to be right dense, and if ρ(r) = r, r is said to be left dense. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2">View MathML</a> has a right scattered minimum m, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M21">View MathML</a>; Otherwise set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M22">View MathML</a>. The backward graininess <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M23">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M24">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2">View MathML</a> has a left scattered maximum M, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M25">View MathML</a>; Otherwise set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M26">View MathML</a>. The forward graininess <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M27">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M28">View MathML</a>

Definition 2.2. [7,22] For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M30">View MathML</a>, we define the "Δ" derivative of x(t), xΔ(t), to be the number (when it exists), with the property that, for any ε > 0, there is neighborhood U of t such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M31">View MathML</a>

for all s U. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M32">View MathML</a>, we define the "∇" derivative of x(t),xΔ (t), to be the number(when it exists), with the property that, for any ε > 0, there is a neighborhood V of t such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M33">View MathML</a>

for all s V.

Definition 2.3. [22] If FΔ (t) = f(t), then we define the "Δ" integral by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M34">View MathML</a>

If F(t) = f(t), then we define the "∇" integral by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M35">View MathML</a>

Lemma 2.1. [23]The following formulas hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M36">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M37">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M38">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M39">View MathML</a>.

Lemma 2.2. [7, Theorem 1.75 in p. 28] If f Crd and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M30">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M40">View MathML</a>

According to [23, Theorem 1.30 in p. 9], we have the following lemma, which can be proved easily. Here, we omit it.

Lemma 2.3. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M41">View MathML</a>and f Cld.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M42">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M43">View MathML</a>

where the integral on the right is the usual Riemann integral from calculus.

(ii) If [a, b] consists of only isolated points, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M44">View MathML</a>

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M45">View MathML</a>, where h > 0, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M46">View MathML</a>

(iv) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M47">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M48">View MathML</a>

In what follows, we list the fixed point theorems that will be used in this article.

Theorem 2.4. [24]Let E be a Banach space and P E be a cone. Suppose 1, Ω2 E open and bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M49">View MathML</a>. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M50">View MathML</a>is completely continuous. If one of the following conditions holds

(i) ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω1 P, ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω2 P;

(ii) ∥Ax∥ ≥ ∥x∥, ∀x ∈ ∂Ω1 P, ∥Ax∥ ≤ ∥x∥, ∀x ∈ ∂Ω2 P.

Then, A has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M51">View MathML</a>.

Theorem 2.5. [25]Let P be a cone in the real Banach space E. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M52">View MathML</a>

If α and γ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ(0) = 0 such that for some positive constants r, M,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M53">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M54">View MathML</a>. Further, suppose there exists positive numbers a < b < r such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M55">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M56">View MathML</a>is completely continuous operator satisfying

(i) γ(Au) > r for all u ∈ ∂P(γ, r);

(ii) θ(Au) < b for all u ∂P(θ, r);

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M57">View MathML</a>and α(Au) > a for all u ∂P(α, a).

Then, A has at least two fixed points u1 and u2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M58">View MathML</a>

Let a, b, c be constants, Pr = {u P : ∥u∥ < r}, P(ψ, b, d) = {u P : a ψ(u), ∥u∥ ≤ b}.

Theorem 2.6. [26]Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M59">View MathML</a>be a completely continuous map and ψ be a nonnegative continuous concave functional on P such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M60">View MathML</a>, there holds ψ(u) ≤ ∥u∥. Suppose there exist a, b, d with 0 < a < b < d c such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M61">View MathML</a>and ψ(Au) > b for all u P(ψ, b, d);

(ii) ∥Au∥ < a for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M62">View MathML</a>;

(iii) ψ(Au) > b for all u P(ψ, b, d) with Au∥ > d.

Then, A has at least three fixed points u1,u2, and u3 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M63">View MathML</a>

Let the Banach space E = Cld[0, T] be endowed with the norm ∥u∥ = supt ∈ [0,T] u(t), and cone P E is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M64">View MathML</a>

It is obvious that ∥u∥ = u(T) for u P. Define A : P E as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M65">View MathML</a>

for t ∈ [0, T].

In what follows, we give the main lemmas which are important for getting the main results.

Lemma 2.7. A : P P is completely continuous.

Proof. First, we try to prove that A : P P.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M66">View MathML</a>

Thus, (Au)Δ (T) = 0 and by Lemma 2.1 we have (Au)Δ∇ (t) = -h(t)f(t, u(t)) ≤ 0 for t ∈ (0, T). Consequently, A : P P.

By standard argument we can prove that A is completely continuous. For more details, see [27]. The proof is complete.

Lemma 2.8. For u P, there holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M67">View MathML</a>for t ∈ [0,T].

Proof. For u P, we have uΔ∇ (t) ≤ 0, it follows that uΔ (t) is non-increasing. Therefore, for 0 < t < T,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M68">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M69">View MathML</a>

(2.2)

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M70">View MathML</a>

(2.3)

Combining (2.1) and (2.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M71">View MathML</a>

as u(0) ≥ 0, it is immediate that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M72">View MathML</a>

The proof is complete.

3 Existence of at least one positive solution

First, we give some notations. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M73">View MathML</a>

Theorem 3.1. Assume in addition to (S1) and (S2), the following conditions are satisfied, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M74">View MathML</a> such that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M75">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M76">View MathML</a>.

Then, BVP (1.6) has at least one positive solution.

Proof. Cone P is defined as above. By Lemma 2.7 we know that A : P P is completely continuous. Set Ωr = {u E, ∥u∥ < r}. In view of (H1), for u r P,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M77">View MathML</a>

which means that for u ∈ ∂Ωr P, ||Au|| ≤ ||u||.

On the other hand, for u P, in view of Lemma 2.8, there holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M78">View MathML</a>, for t ∈ [ξ1, T]. Denote Ωρ = {u E, ∥u∥ < ρ}. Then for u ∈ ∂Ωρ P, considering (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M79">View MathML</a>

which implies that for u ρP, ∥Au∥ ≥ ∥u∥ Therefore, the immediate result of Theorem 2.4 is that A has at least one fixed point u ∈ (Ωρ\Ωr) ∩ P. Also, it is obvious that the fixed point of A in cone P is equivalent to the positive solution of BVP (1.6), this yields that BVP (1.6) has at least one positive solution u satisfies r ≤ ∥u∥ ≤ ρ. The proof is complete.

Here is an example.

Example 3.2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M80">View MathML</a>. Consider the following four point BVP on time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M81">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M82">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M83">View MathML</a>

and h(t) = 1, T = 4, ξ1 = 2, ξ2 = 3, δ = 2, β1 = β2 = 1,p = q = 2. In what follows, we try to calculate Λ, B. By Lemmas 2.2 and 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M84">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M85">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M86">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M87">View MathML</a>. Then, we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M88">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M89">View MathML</a>

Thus, if all the conditions in Theorem 3.1 satisfied, then BVP (3.1) has at least one positive solution lies between 100 and 1000.

4 Existence of at least two positive solutions

In this section, we will apply fixed point Theorem 2.5 to prove the existence of at least two positive solutions to the nonlinear BVP (1.6).

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M90">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M91">View MathML</a>

and define the increasing, nonnegative, continuous functionals γ, θ,α on P by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M92">View MathML</a>

We can see that, for u ∈ P, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M93">View MathML</a>

In addition, Lemma 2.8 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M94">View MathML</a> which means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M95">View MathML</a>

We also see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M96">View MathML</a>

For convenience, we give some notations,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M97">View MathML</a>

Theorem 4.1. Assume in addition to (S1), (S2) there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M98">View MathML</a>such that the following conditions hold

(H3) f(t, u) > ϕp(c/M) for t ∈ [ξ1,T] u ∈ [c,Tc/ξ1];

(H4) f(t, u) < ϕp(b/K) for t ∈ [0,ξm-2], u ∈ [b,Tb/ξm-2];

(H5) f(t, u) > ϕp(a/L) for t ∈ [η,T], u ∈ [a,Ta/η].

Then BVP (1.6) has at least two positive solutions u1 and u2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M99">View MathML</a>

(4.1)

Proof. From Lemma 2.7 we know that A : P(γ, c) → P is completely continuous. In what follows, we will prove the result step by step.

Step one: To verify (i) of theorem 2.5 holds.

We choose u ∈ ∂P(γ,c), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M100">View MathML</a>. This implies that u(t) ≥ c for t ∈ [ξ1,T], considering that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M101">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M102">View MathML</a>

As a consequence of (H3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M103">View MathML</a>

Since Au P, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M104">View MathML</a>

Thus, (i) of Theorem 2.5 is satisfied.

Step two: To verify (ii) of Theorem 2.5 holds.

Let u ∂P(θ,b), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M105">View MathML</a>, this implies that 0 ≤ u(t) ≤ b, t ∈ [0,ξm-2] and since u P, we have ∥u∥ = u(T), note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M106">View MathML</a>. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M107">View MathML</a>

From (H4) we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M108">View MathML</a> for t ∈ [0, ξm-2] and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M109">View MathML</a>

Thus, (ii) of Theorem 2.5 holds.

Step three: To verify (iii) of Theorem 2.5 holds.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M110">View MathML</a>, obviously, u0(t) ∈ P(α, a) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M111">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M112">View MathML</a>.

Now, let u ∂P(α, a), then, α(u) = mint∈[η,T] u(t) = u(η) = a. Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M113">View MathML</a>. Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M114">View MathML</a>

From assumption (H5) we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M115">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M116">View MathML</a>

Therefore, all the conditions of Theorem 2.5 are satisfied, thus A has at least two fixed points in P(γ,c), which implies that BVP (1.6) has at least two positive solutions u1,u2 which satisfies (4.1). The proof is complete.

Example 4.2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M117">View MathML</a>. Consider the following four point boundary value problem on time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M118">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M119">View MathML</a>

and h(t) = t, T = 8, ξ1 = 1, ξ2 = 2, δ = 1, β1 = 1, β2 = 2,p = 3/2, q = 3. In what follows, we try to calculate K, M, L. By Lemmas 2.2 and 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M120">View MathML</a>

Let a = 106, b = 108, c = 109, then we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M121">View MathML</a>, for t ∈ [1, 8], u ∈ [109, 8 × 109];

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M122">View MathML</a>, for t ∈ [0, 2], u ∈ [108, 4 × 108];

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M123">View MathML</a>, for t ∈ [4, 8], u ∈ [106, 2 × 106].

Thus, if all the conditions in Theorem 4.1 are satisfied, then BVP (4.2) has at least two positive solutions satisfying (4.1).

5 Existence of at least three positive solutions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M124">View MathML</a>, then 0 < ψ(u) ≤ ∥u∥. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M125">View MathML</a>

In this section, we will use fixed point Theorem 2.6 to get the existence of at least three positive solutions.

Theorem 5.1. Assume that there exists positive number d, ν, g satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M126">View MathML</a>, such that the following conditions hold.

(H6) f(t, u) < ϕp(d/R), t ∈ [0,T],u ∈ [0,d];

(H7) f(t, u) > ϕp(ν/D), t ∈ [ξ1, T], u ∈ [ν, Tυ/ξ1];

(H8) f(t, u) ≤ ϕp(g/R), t ∈ [0,T],u ∈ [0,g],

then BVP (1.6) has at least three positive solutions u1, u2, u3 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M127">View MathML</a>

(5.1)

Proof. From Lemma 2.8 we know that A : P P is completely continuous. Now we only need to show that all the conditions in Theorem 2.6 are satisfied.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M128">View MathML</a>. By (H8), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M129">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M130">View MathML</a>. Similarly, by (H6), we can prove (ii) of Theorem 2.6 is satisfied.

In what follows, we try to prove that (i) of theorem 2.6 holds. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M131">View MathML</a>, obviously, ψ(u1) > ν, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M132">View MathML</a>. For u P(ψ,ν,Tν/ξ1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M133">View MathML</a>

It remains to prove (iii) of Theorem 2.6 holds. For u P(ψ, ν, Tυ/ξ1), with ∥Au∥ > /ξ1, in view of Lemma 2.8, there holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M134">View MathML</a>, which implies that (iii) of Theorem 2.6 holds.

Therefore, all the conditions in Theorem 2.6 are satisfied. Thus, BVP (1.6) has at least three positive solutions satisfying (5.1). The proof is complete.

Example 5.2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M135">View MathML</a>. Consider the following four point boundary value problem on time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M2">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M136">View MathML</a>

(5.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M137">View MathML</a>

and h(t) = et, T = 2, ξ1 = 1/2, ξ2 = 1, δ = 3, β1 = 2, β2 = 3, p = 4, q = 4/3. In what follows, we try to calculate D, R. By Lemmas 2.2 and 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M138">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/4/mathml/M139">View MathML</a>

Let d = 40, ν = 50, g = 400, then we have

(i) f(t, u) < 7.027 = (40/20.8832)3 = ϕp(d/R), for t ∈ [0, 2], u ∈ [0, 40];

(ii) f(t, u) > 23.2375 = (50/17.5216)3 = ϕp(ν/D), for t ∈ [1/2, 2], u ∈ [50, 200];

(iii) f(t, u) < 7027.305 = (400/20.8832)3 = ϕp(g/R), for t ∈ [0, 2], u ∈ [0, 400].

Thus, if all the conditions in Theorem 5.1 are satisfied, then BVP (5.2) has at least three positive solutions satisfying (5.1).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

WG and HL conceived of the study, and participated in its coordination. JZ drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors were very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The study was supported by Pre-research project and Excellent Teachers project of the Fundamental Research Funds for the Central Universities (2011YYL079, 2011YXL047).

References

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

  2. Agarwal, RP, Bohner, M, Li, WT: Nonoscillation and Oscillation Theory for Functional Differential Equations. Pure and Applied Mathematics Series, Dekker, FL (2004)

  3. Moshinsky, M: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol Soc Mat Mexicana. 7, 10–25 (1950)

  4. Timoshenko, S: Theory of Elastic Stability. McGraw-Hill, New York (1961)

  5. Zou, Y, Hu, Q, Zhang, R: On numerical studies of multi-point boundary value problem and its fold bifurcation. Appl Math Comput. 185, 527–537 (2007). Publisher Full Text OpenURL

  6. Il'in, VA, Moiseev, EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Diff Equ. 23, 979–987 (1987)

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

  8. Agarwal, RP, O'Regan, D: Nonlinear boundary value problems on time scales. Nonlinear Anal. 44, 527–535 (2001). Publisher Full Text OpenURL

  9. Anderson, D: Solutions to second-order three-point problems on time scales. J Diff Equ Appl. 8, 673–688 (2002). Publisher Full Text OpenURL

  10. Kaufmann, ER: Positive solutions of a three-point boundary value problem on a time scale. Electron J Diff Equ. 82, 1–11 (2003)

  11. Chyan, CJ, Henderson, J: Twin solutions of boundary value problems for differential equations on measure chains. J Comput Appl Math. 141, 123–131 (2002). Publisher Full Text OpenURL

  12. Agarwal, RP, Lü, HSh, O'Regan, D: Eigenvalues and the one-dimensional p-Laplacian. J Math Anal Appl. 266, 383–400 (2002). Publisher Full Text OpenURL

  13. Sun, H, Tang, L, Wang, Y: Eigenvalue problem for p-Laplacian three-point boundary value problem on time scales. J Math Anal Appl. 331, 248–262 (2007). Publisher Full Text OpenURL

  14. Geng, F, Zhu, D: Multiple results of p-Laplacian dynamic equations on time scales. Appl Math Comput. 193, 311–320 (2007). Publisher Full Text OpenURL

  15. He, Z, Jiang, X: Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales. J Math Anal Appl. 321, 911–920 (2006). Publisher Full Text OpenURL

  16. Hong, S: Triple positive solutions of three-point boundary value problems for p-Laplacian dynamic equations. J Comput Appl Math. 206, 967–976 (2007). Publisher Full Text OpenURL

  17. Graef, J, Kong, L: First-order singular boundary value problems with p-Laplacian on time scales. J Diff Equ Appl. 17, 831–839 (2011). Publisher Full Text OpenURL

  18. Anderson, DR: Existence of solutions for a first-order p-Laplacian BVP on time scales. Nonlinear Anal. 69, 4521–4525 (2008). Publisher Full Text OpenURL

  19. Goodrich, CS: Existence of a positive solution to a first-order p-Laplacian BVP on a time scale. Nonlinear Anal. 74, 1926–1936 (2011). Publisher Full Text OpenURL

  20. He, ZM: Double positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales. Appl Anal. 84, 377–390 (2005). Publisher Full Text OpenURL

  21. Anderson, DR: Twin n-point boundary value problem. Appl Math Lett. 17, 1053–1059 (2004). Publisher Full Text OpenURL

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

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

  24. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

  25. Avery, RI, Henderson, J: Two positive fixed points of nonlinear operators on ordered Banach spaces. Comm Appl Nonlinear Anal. 8, 27–36 (2001)

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

  27. Zhao, J: Nonlocal boundary value problems of ordinary differential equations and dynamical equations on time scales. Doctoral thesis, Beijing Institute of Technology (2009)