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Monotone and convex positive solutions for fourth-order multi-point boundary value problems

Yang Liu12*, Zhang Weiguo1 and Shen Chunfang2

Author affiliations

1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China

2 Department of Mathematics, Hefei Normal University, Hefei, Anhui Province 230061, PR China

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Citation and License

Boundary Value Problems 2011, 2011:21  doi:10.1186/1687-2770-2011-21

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/21


Received:27 December 2010
Accepted:5 September 2011
Published:5 September 2011

© 2011 Liu 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

The existence results of multiple monotone and convex positive solutions for some fourth-order multi-point boundary value problems are established. The nonlinearities in the problems studied depend on all order derivatives. The analysis relies on a fixed point theorem in a cone. The explicit expressions and properties of associated Green's functions are also given.

MSC: 34B10; 34B15.

Keywords:
multi-point boundary value problem; positive solution; cone; fixed point

1 Introduction

Boundary value problems for second and higher order nonlinear differential equations play a very important role in both theory and applications. For example, the deformations of an elastic beam in the equilibrium state can be described as a boundary value problem of some fourth-order differential equations. Owing to its importance in application, the existence of positive solutions for nonlinear second and higher order boundary value problems has been studied by many authors. We refer to recent contributions of Ma [1-3], He and Ge [4], Guo and Ge [5], Avery et al. [6,7], Henderson [8], Eloe and Henderson [9], Yang et al. [10], Webb and Infante [11,12], and Agarwal and O'Regan [13]. For survey of known results and additional references, we refer the reader to the monographs by Agarwal [14] and Agarwal et al. [15].

When it comes to positive solutions for nonlinear fourth-order ordinary differential equations, two point boundary value problems are studied extensively, see [16-24]. Few papers deal with the multi-point cases. Furthermore, for nonlinear fourth-order equations, only the situation that the nonlinear term does not depend on the first, second and third order derivatives are considered, see [16-23]. Few paper deals with the situation that lower order derivatives are involved in the nonlinear term explicitly. In fact, the derivatives are of great importance in the problem in some cases. For example, in the linear elastic beam equation (Euler-Bernoulli equation)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M1">View MathML</a>

where u(t) is the deformation function, L is the length of the beam, f(t) is the load density, E is the Young's modulus of elasticity and I is the moment of inertia of the cross-section of the beam. In this problem, the physical meaning of the derivatives of the function u(t) is as follows: u(4)(t) is the load density stiffness, u'''(t) is the shear force stiffness, u''(t) is the bending moment stiffness and the u'(t) is the slope. If the payload depends on the shear force stiffness, bending moment stiffness or the slope, the derivatives of the unknown function are involved in the nonlinear term explicitly.

In this paper, we are interested in the positive solution for fourth-order nonlinear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M2">View MathML</a>

(1.1)

subject to multi-point boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M3">View MathML</a>

(1.2)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M4">View MathML</a>

(1.3)

where 0 < ξ1 < ξ2 < ⋯ < ξm-2 < 1, βi > 0, 1 = 1, 2, ..., m - 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M5">View MathML</a>, and f C([0, 1] × R4, [0, +∞)).

One can see that all lower order derivatives are involved in the nonlinear term explicitly and the BCs are the m-point cases. In this sense, the problems studied in this paper are more general than before. In the paper, multiple monotone and convex positive solutions for problems (1.1), (1.2) and (1.1), (1.3) are established. The results presented extend the study for fourth-order boundary value problems of nonlinear ordinary differential equations.

This paper is organized as follows. In Section 2, we present some preliminaries and lemmas. Section 3 is devoted to the existence of at least three convex and increasing positive solutions for problem (1.1), (1.2). In Section 4, we prove that there exist at least three convex and decreasing positive solutions for problem (1.1), (1.3).

2 Preliminaries and lemmas

In this section, some preliminaries and lemmas used later are presented.

Definition 2.1 The map α is said to be a nonnegative continuous convex functional on cone P of a real Banach space E provided that α : P → [0, +∞) is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M6">View MathML</a>

Definition 2.2 The map β is said to be a nonnegative continuous concave functional on cone P of a real Banach space E provided that β : P → [0, +∞) is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M7">View MathML</a>

Let γ, θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional on P and ψ be a nonnegative continuous functional on P. Then for positive numbers a, b, c and d, we define the following convex sets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M8">View MathML</a>

and a closed set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M9">View MathML</a>

Lemma 2.1 [25] Let P be a cone in Banach space E. Let γ, θ be nonnegative continuous convex functionals on P, α be a nonnegative continuous concave functional and ψ be a nonnegative continuous functional on P satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M10">View MathML</a>

such that for some positive numbers l and d,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M11">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M12">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M13">View MathML</a> is completely continuous and there exist positive numbers a, b, c with a < b such that

(S1) {x P (γ, θ, α, b, c, d)|α(x) > b} ≠ ∅ and α(Tx) > b for x P (γ, θ, α, b, c, d);

(S2) α(Tx) > b for x P (γ, α, b, d) with θ(Tx) > c;

(S3) 0 ∉ R(γ, ψ, a, d) and ψ(Tx) < a for x R(γ, ψ, a, d) with ψ(x) = a.

Then T has at least three fixed points x1, x2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M14">View MathML</a> such that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M15">View MathML</a>

3 Positive solutions for problem (1.1), (1.2)

We begin with the fourth-order m-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M16">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M17">View MathML</a>

(3.2)

where 0 < ξ1 < ξ2 < ⋯ < ξm-2 < 1, βi > 0, i = 1, 2, ..., m - 2.

The following assumption will stand throughout this section:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M18">View MathML</a>

Lemma 3.1 Denote ξ0 = 0, ξm -1 = 1, β0 = βm -1 = 0, and y(t) ∈ C[0, 1]. Problem (3.1), (3.2) has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M19">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M20">View MathML</a>

for ξi-1 s ξi, i = 1, 2, ..., m -1.

Proof Let G(t, s) be the Green's function of problem x(4)(t) = 0 with boundary condition (3.2). We can suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M21">View MathML</a>

Considering the definition and properties of Green's function together with the boundary condition (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M22">View MathML</a>

A straightforward calculation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M23">View MathML</a>

These give the explicit expression of the Green's function and the proof of Lemma 3.1 is completed.

Lemma 3.2 One can see that G(t, s) ≥ 0, t, s ∈ [0, 1].

Proof For ξi-1 s ξi, i = 1, 2, ..., m - 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M24">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M25">View MathML</a>, 0 ≤ t, s ≤ 1. Thus G(t, s) is increasing on t. By a simple computation, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M26">View MathML</a>

These ensures that G(t, s) ≥ 0, t, s ∈ [0, 1].

Lemma 3.3 Suppose x(t) ∈ C3[0, 1] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M27">View MathML</a>

Furthermore x(4)(t) ≥ 0 and there exist t0 such that x(4)(t0) > 0. Then x(t) has the following properties:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M28">View MathML</a>

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M29">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M31">View MathML</a> are positive constants.

Proof Since x(4)(t) ≥ 0, t ∈ [0, 1], then x'''(t) is increasing on [0, 1]. Considering x'''(1) = 0, we have x'''(t) ≤ 0, t ∈ [0, 1]. Thus x''(t) is decreasing on [0, 1]. Considering this together with the boundary condition x''(1) = 0, we conclude that x''(t) ≥ 0. Then x(t) is convex on [0, 1]. Taking into account that x'(0) = 0, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M32">View MathML</a>

(1) From the concavity of x(t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M33">View MathML</a>

Multiplying both sides with βi and considering the boundary condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M34">View MathML</a>

(3.3)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M35">View MathML</a>

(2) Considering the mean-value theorem together with the concavity of x(t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M36">View MathML</a>

(3.4)

Multiplying both sides with βi and considering the boundary condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M37">View MathML</a>

(3.5)

which yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M38">View MathML</a>.

(3) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M39">View MathML</a> and x'(0) = 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M40">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M41">View MathML</a> and x"(1) = 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M42">View MathML</a>

Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M43">View MathML</a>

These give the proof of Lemma 3.3.

Remark Lemma 3.3 ensures that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M44">View MathML</a>

Let Banach space E = C3[0, 1] be endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M45">View MathML</a>

Define the cone P E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M46">View MathML</a>

Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionals γ, θ and the nonnegative continuous functional ψ be defined on the cone by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M47">View MathML</a>

By Lemma 3.3, the functionals defined above satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M48">View MathML</a>

(3.6)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M49">View MathML</a>

Assume that there exist constants 0 < a, b, d with a < b < λd such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M50">View MathML</a>

Theorem 3.1 Under assumptions (A1)-(A3), problem (1.1), (1.2) has at least three positive solutions x1, x2, x3 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M51">View MathML</a>

Proof Problem (1.1, 1.2) has a solution x = x(t) if and only if x solves the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M52">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M53">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M12">View MathML</a>, considering Lemma 3.3 and assumption (A1), we have f(t, x(t), x'(t), x''(t), x'''(t)) ≤ d. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M54">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M13">View MathML</a>. An application of the Arzela-Ascoli theorem yields that T is a completely continuous operator. The fact that the constant function x(t) = b/δ P(γ, θ, α, b, c, d) and α(b/δ) > b implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M55">View MathML</a>

For x P(γ, θ, α, b, c, d), we have b x(t) ≤ b/δ and |x'''(t)| < d. From assumption (A2), we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M56">View MathML</a>

Hence, by definition of α and the cone P, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M57">View MathML</a>

which means α(Tx) > b, ∀x P(γ, θ, α, b, b/δ, d). This ensures that condition (S1) of Lemma 2.1 is fulfilled.

Second, with (3.4) and b < λd, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M58">View MathML</a>

for all x P(γ, α, b, d) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M59">View MathML</a>.

Thus, condition (S2) of Lemma 2.1 holds. Finally we show that (S3) also holds. We see ψ(0) = 0 < a and 0 ∉ R(γ, ψ, a, d). Suppose that x R(γ, ψ, a, d) with ψ(x) = a, then by the assumption of (A3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M60">View MathML</a>

which ensures that condition (S3) of Lemma 2.1 is fulfilled. Thus, an application of Lemma 2.1 implies that the fourth-order m-point boundary value problem (1.1, 1.2) has at least three positive convex increasing solutions x1, x2, x3 with the properties that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M61">View MathML</a>

4 Positive solutions for problem (1.1), (1.3)

The following assumption will stand throughout this section:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M62">View MathML</a>

Lemma 4.1 Denote ξ0 = 0, ξm-1 = 1, β0 = βm-1 = 0, the Green's function of problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M63">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M64">View MathML</a>

(4.2)

is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M65">View MathML</a>

for i = 1, 2, ..., m - 1.

Proof Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M66">View MathML</a>

Considering the definition and properties of Green's function together with the boundary condition (4.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M67">View MathML</a>

Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M68">View MathML</a>

The proof of Lemma 4.1 is completed.

Lemma 4.2 One can see that H(t, s) ≥ 0, t, s ∈ [0, 1].

Proof For ξi-1 s ξi, i = 1, 2, ..., m - 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M69">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M70">View MathML</a>, 0 ≤ t, s ≤ 1, which implies that H(t, s) is decreasing on t. The fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M71">View MathML</a>

ensures that H(t, s) ≥ 0, t, s ∈ [0, 1].

Lemma 4.3 If x(t) ∈ C3[0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M72">View MathML</a>

and x(4)(t) ≥ 0, there exists t0 such that x(4)(t0) > 0, then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M73">View MathML</a>

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M74">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M75">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M76">View MathML</a> are positive constants.

Proof It follows from the same methods as Lemma 3.3 that x(t) is convex on [0, 1]. Taking into account that x'(1) = 0, one can see that x(t) is decreasing on [0, 1] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M77">View MathML</a>

(1) From the concavity of x(t), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M78">View MathML</a>

Multiplying both sides with βi and considering the boundary condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M79">View MathML</a>

(4.3)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M80">View MathML</a>

(2) Considering the mean-value theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M81">View MathML</a>

From the concavity of x similarly with above we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M82">View MathML</a>

(4.4)

Considering (4.3) together with (4.4) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M83">View MathML</a>.

(3) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M84">View MathML</a> and x'(1) = 0, x''(1) = 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M85">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M86">View MathML</a>

Remark We see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M87">View MathML</a>

Let Banach space E = C3[0, 1] be endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M88">View MathML</a>

Define the cone P E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M89">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M90">View MathML</a>

Assume that there exist constants 0 < a, b, d with a < b < λ1d such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M91">View MathML</a>

Theorem 4.1 Under assumptions (A4)-(A6), problem (1.1), (1.3) has at least three positive solutions x1, x2, x3 with the properties that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M92">View MathML</a>

Proof Problem (1.1), (1.3) has a solution x = x(t) if and only if x solves the operator equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M93">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M94">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M95">View MathML</a>, considering Lemma 4.3 and assumption (A4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M96">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M97">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M98">View MathML</a> and T1 is a completely continuous operator obviously. The fact

that the constant function x(t) = b/δ1 P1(γ, θ, α, b, c, d) and α(b/δ1) > b implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M99">View MathML</a>

This ensures that condition (S1) of Lemma 2.1 holds.

For x P1(γ, θ, α, b, c, d), we have b x(t) ≤ b/δ1 and |x'''(t)| < d. From assumption (A4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M100">View MathML</a>

Hence, by definition of α and the cone P1, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M101">View MathML</a>

which means α(T1x) > b, ∀x P1(γ, θ, α, b, b/δ, d).

Second, with (4.4) and b < λ1d, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M102">View MathML</a>

for all x P1(γ, α, b, d) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M103">View MathML</a>.

Thus, condition (S2) of Lemma 2.1 holds. Finally we show that (S3) also holds. We see ψ(0) = 0 < a and 0 ∉ R(γ, ψ, a, d). Suppose that x R(γ, ψ, a, d) with ψ(x) = a, then by the assumption of (A6),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M104">View MathML</a>

which ensures that condition (S3) of Lemma 2.1 is satisfied. Thus, an application of Lemma 2.1 implies that the fourth-order m-point boundary value problem (1.1), (1.3) has at least three positive convex decreasing solutions x1, x2, x3 satisfying the conditions that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/21/mathml/M105">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

We are indebted to the anonymous referee for a detailed reading and useful comments and suggestions, which allowed us to improve this work. This work was supported by the Anhui Provincial Natural Science Foundation (10040606Q50), National Natural Science Foundation of China (No.11071164), Shanghai Natural Science Foundation (No.10ZR1420800).

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