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Existence of solutions for integral boundary value problems of second-order ordinary differential equations

Hongyu Li1* and Fei Sun2

Author Affiliations

1 Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao, Shandong, 266590, P.R. China

2 Jiamusi Hongqi Middle School, Jiamusi, Heilongjiang, 154011, P.R. China

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Boundary Value Problems 2012, 2012:147  doi:10.1186/1687-2770-2012-147

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/147


Received:24 June 2012
Accepted:26 November 2012
Published:17 December 2012

© 2012 Li and Sun; 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 investigate the existence of solutions for some second-order integral boundary value problems, by applying new fixed point theorems in Banach spaces with the lattice structure derived by Sun and Liu.

MSC: 34B15, 34B18, 47H11.

Keywords:
lattice; fixed point; integral boundary value problem; sign-changing solution

1 Introduction

In this paper, we consider the following second-order integral boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M3">View MathML</a> is nonnegative with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M4">View MathML</a>.

The multi-point boundary value problems for ordinary differential equations have been well studied, especially on a compact interval. For example, the study of three-point boundary value problems for nonlinear second-order ordinary differential equations was initiated by Gupta (see [1]). Since then, the existence of solutions for nonlinear multi-point boundary value problems has received much attention from some authors; see [2-6] for reference.

The integral boundary value problems of ordinary differential equations arise in different areas of applied mathematics and physics such as heat conduction, underground water flow, thermo-elasticity and plasma physics (see [7,8] and the references therein). Moreover, boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two, three, multi-point and integral boundary value problems as special cases (see [9,10]). For boundary value problems with other integral boundary conditions, we refer the reader to the papers [11-21] and the references therein.

In [15], Zhang and Sun studied the following differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M5">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M3">View MathML</a> is nonnegative with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M4">View MathML</a>. By fixed-point index theory, the existence and multiplicity of sign-changing solutions was discussed.

As we know, nearly all the methods computing the topological degree depend on cone mappings. Recently, Sun and Liu introduced some new computation of topological degree when the concerned operators are not cone mappings in ordered Banach spaces with the lattice structure (for details, see [22-25]). To the best of our knowledge, there is only one paper to use this new computation of topological degree to study an integral boundary value problem with the asymptotically nonlinear term (see [16]).

Motivated by [15,16,22-25], this paper is concerned with the boundary value problem (1.1) under sublinear conditions. The method we use is based on some recent fixed point theorems derived by Sun and Liu [22,23], which are different from [16] and the results we obtain are different from [11-21].

This paper is organized as follows. In Section 2, we recall some properties of the lattice, new fixed point theorems and some lemmas that will be used to prove the main results. In Section 3, we prove the main results and, finally, we give concrete examples to illustrate the applicability of our theory.

2 Preliminaries

We first give some properties of the lattice and give new fixed point theorems with the lattice structure (see [22-25]).

Let E be a Banach space with a cone P. Then E becomes an ordered Banach space under the partial ordering ≤ which is induced by P. P is said to be normal if there exists a positive constant N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M9">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M10">View MathML</a>. P is called solid if it contains interior points, i.e., int <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M11">View MathML</a>.

Definition 2.1[22-25]

We call E a lattice under the partial ordering ≤ if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M13">View MathML</a> exist for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M14">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M15">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M16">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M18">View MathML</a> are called the positive part and the negative part of x, respectively, and obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M19">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M21">View MathML</a>. For convenience, we use the notations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M23">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M24">View MathML</a> be a bounded linear operator. B is said to be positive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M25">View MathML</a>.

Definition 2.2[22-25]

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M27">View MathML</a> be a nonlinear operator. A is said to be quasi-additive on lattice if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M28">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M29">View MathML</a>

Let P be a cone of a Banach space E. x is said to be a positive fixed point of A if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M30">View MathML</a> is a fixed point of A; x is said to be a negative fixed point of A if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M31">View MathML</a> is a fixed point of A; x is said to be a sign-changing fixed point of A if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M32">View MathML</a> is a fixed point of A.

Lemma 2.1[22,23]

LetPbe a normal cone ofE, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M33">View MathML</a>be completely continuous and quasi-additive on lattice. Suppose that

(i) there exist a positive bounded linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M36">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M37">View MathML</a>

(ii) there exist a positive bounded linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M39">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M40">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M42">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M43">View MathML</a>is the spectral radius of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M44">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M45">View MathML</a>);

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M46">View MathML</a>, the Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>ofAatθexists, and 1 is not an eigenvalue of the linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>, the sumμof the algebraic multiplicities for all eigenvalues of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>lying in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M50">View MathML</a>is an odd number.

Then the operatorAhas at least one nonzero fixed point.

Lemma 2.2[22]

Let the conditions (i), (ii) and (iii) of Lemma 2.1 be satisfied. Suppose, in addition, that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M51">View MathML</a>; the Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>ofAatθexists; 1 is not an eigenvalue of the linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>; the sumμof the algebraic multiplicities for all eigenvalues of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>lying in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M55">View MathML</a>is an even number and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M56">View MathML</a>

Then the operatorAhas at least three fixed points: one positive fixed point, one negative fixed point and one sign-changing fixed point.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M57">View MathML</a> with the normal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M58">View MathML</a>, then E is a Banach space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M59">View MathML</a>, then P is a cone of E. It is easy to see that E is a lattice under the partial ordering ≤ that is induced by P.

For convenience, list the following condition.

(H0)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M60">View MathML</a>

is the sequence of positive solutions of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M61">View MathML</a>.

Define the operators A, B and F:

(2.1)

(2.2)

(2.3)

where

It is obvious that the fixed points of the operator A defined by (2.3) are the solutions of the boundary value problem (1.1) (see [15,16]).

Lemma 2.3[16]

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M24">View MathML</a>is a completely continuous linear operator;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M33">View MathML</a>is a completely continuous operator;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M68">View MathML</a>is quasi-additive on the lattice;

(iv) the eigenvalues of the linear operatorBare<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M69">View MathML</a>and the algebraic multiplicity of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M70">View MathML</a>is equal to 1, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M71">View MathML</a>is defined by (H0);

(v) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M72">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M73">View MathML</a>is the spectral radius of the operatorB.

3 Main results

Let us list some conditions for convenience.

(H1) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M74">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M75">View MathML</a>

(H2) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M77">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M78">View MathML</a> is defined by (H0).

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M79">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M80">View MathML</a>.

(H4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M81">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M84">View MathML</a> is defined by (H0).

Theorem 3.1Suppose that (H0), (H1), (H2), (H3), (H4) are satisfied andnis an odd number in (H4). Then the boundary value problem (1.1) has at least a nontrivial solution.

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M85">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M86">View MathML</a>. By (H2), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M87">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M88">View MathML</a>

(3.1)

So, by (3.1) and (H1), we know that

(3.2)

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M91">View MathML</a>.

By (3.2) and (3.3), we have

(3.4)

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M97">View MathML</a>, B is defined by (2.1). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M100">View MathML</a> is a positive completely continuous operator. By Lemma 2.3, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M101">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M102">View MathML</a>.

By (H3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M46">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M104">View MathML</a>

(3.6)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M105">View MathML</a>. By Lemma 2.3, 1 is not an eigenvalue of the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M107">View MathML</a>, n is an odd number, the sum of the algebraic multiplicities for all eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a> lying in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M50">View MathML</a> is an odd number. By Lemma 2.1, the operator A has at least one nonzero fixed point. So, the boundary value problem (1.1) has at least one nontrivial solution. □

Theorem 3.2Suppose (H0), (H2), (H3), (H4) are satisfied andnis an even number in (H4). In addition, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M110">View MathML</a>

(3.7)

Then the boundary value problem (1.1) has at least three nontrivial solutions: one positive solution, one negative solution and one sign-changing solution.

Proof By (3.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M111">View MathML</a>

(3.8)

By (3.1) and (3.8), (3.4) and (3.5) hold. From (H3), (3.6) holds, and by Lemma 2.3, 1 is not an eigenvalue of the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M82">View MathML</a>, n is an even number, the sum of the algebraic multiplicities for all eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M47">View MathML</a> lying in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M55">View MathML</a> is an even number.

Obviously, from (3.8) and (2.2), we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M116">View MathML</a>

(3.9)

From (2.1), we easily know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M118">View MathML</a>.

So, by (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M119">View MathML</a>

By Lemma 2.2, the boundary value problem (1.1) has at least three nontrivial solutions containing a positive solution, a negative solution and a sign-changing solution. □

Remark By Theorem 3.1 and Theorem 3.2, we can see that the methods used in this paper are different from [11-21], and the results are different from [11-21].

Example 3.1 We consider the following integral boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M120">View MathML</a>

(3.10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M121">View MathML</a>

By simple calculations, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M125">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M126">View MathML</a>. It is easy to know that the nonlinear term f satisfies (H1), (H2), (H3), (H4). Thus, the boundary value problem (3.10) has at least a nontrivial solution by Theorem 3.1.

Example 3.2 We consider the following integral boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M127">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M128">View MathML</a>

By simple calculations, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M132">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M133">View MathML</a>. It is easy to know that the nonlinear term f satisfies (H2), (H3), (H4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/147/mathml/M136">View MathML</a>. The boundary value problem (3.11) has at least three nontrivial solutions containing a positive solution, a negative solution and a sign-changing solution by Theorem 3.2.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the reviewers for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of P.R. China (10971179), Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2012SF022, BS2010SF023), Natural Science Foundation of Shandong Province (ZR2010AM035) and SDUST CISE Research Fund.

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