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Infinitely many solutions to superlinear second order m-point boundary value problems

Ruyun Ma1*, Chenghua Gao1 and Xiaoqiang Chen2

Author Affiliations

1 Department of Mathematics, Northwest Normal University, Lanzhou, 730070, PR China

2 School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, PR China

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Boundary Value Problems 2011, 2011:14  doi:10.1186/1687-2770-2011-14

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/14


Received:28 April 2011
Accepted:15 August 2011
Published:15 August 2011

© 2011 Ma 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

We consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M1">View MathML</a>

where:

(1) m ≥ 3, ηi ∈ (0, 1) and αi > 0 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M2">View MathML</a>;

(2) g : ℝ → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M3">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M4">View MathML</a>

(3) p : [0, 1] × ℝ2 → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M5">View MathML</a>

for some C > 0 and β ∈ (0, 1/2).

We obtain infinitely many solutions having specified nodal properties by the bifurcation techniques.

MSC(2000). 34B15, 58E05, 47J10

Keywords:
Nodal solutions; Second order equations; Multi-point boundary value problems; Bifurcation

1 Introduction

We consider the nonlinear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M6">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M7">View MathML</a>

(1.2)

where

(H1) m ≥ 3, ηi ∈ (0, 1) and αi > 0 with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M8">View MathML</a>

(H2) g : ℝ → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M9">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M10">View MathML</a>

(1.4)

(H3) p : [0, 1] × ℝ2 → ℝ is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M11">View MathML</a>

(1.5)

for some C > 0 and β ∈ (0, 1/2).

In order to state our results, we first recall some standard notations to describe the nodal properties of solutions. For any integer, n ≥ 0, Cn[0, 1] will denote the usual Banach space of n-times continuously differentiable functions on [0, 1], with the usual sup-type norm, denoted by || · ||n. Let X := {u C2[0, 1]: u satisfies (1.2)}, Y := C0[0, 1], with the norms | · |2 and | · |0, respectively. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M12">View MathML</a>

with the norms | · |E.

We define a linear operator L : X Y by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M13">View MathML</a>

(1.6)

In addition, for any continuous function g : ℝ → ℝ and any u Y, let g(u) ∈ Y denote the function g(u(x)), x ∈ [0, 1].

Next, we state some notations to describe the nodal properties of solutions of (1.1), see [1] for the details. For any C1 function u, if u(x0) = 0, then x0 is a simple zero of u, if u'(x0) ≠ 0. Now, for any integer k ≥ 1 and any ν ∈ {+, -}, we define sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M14">View MathML</a> consisting of the set of functions u C2[0, 1] satisfying the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M15">View MathML</a>

(i) u(0) = 0, νu'(0) > 0; (ii) u has only simple zeros in [0, 1] and has exactly k - 1 zeros in (0, 1).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M16">View MathML</a>

(i) u(0) = 0, νu'(0) > 0; (ii) u' has only simple zeros in (0, 1) and has exactly k such zeros; (iii) u has a zero strictly between each two consecutive zeros of u'.

Remark 1.1 If we add the restriction u' (1) ≠ 0 on the functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17">View MathML</a> becomes the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18">View MathML</a>, which used in [1]. The reason we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M17">View MathML</a> rather than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18">View MathML</a> is that the Equation (1.1) is not autonomous anymore.

In [1, Remarks 2.1 and 2.2], Rynne pointed out that

a. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M19">View MathML</a>, then u has exactly one zero between each two consecutive zeros of u', and all zeros of u are simple. Thus, u has at least k - 1 zeros in (0, 1), and at most k zeros in (0, 1];

b. The sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18">View MathML</a> are open in X and disjoint;

c. When considering the multi-point boundary condition (1.2), the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M18">View MathML</a> are in fact more appropriate than the sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M20">View MathML</a>.

The main result of this paper is the following

Theorem 1.1 Let (H1)-(H3) hold. Then there exists an integer k0 ≥ 1 such that for all integers k k0 and each ν ∈ {+, -} the problem (1.1), (1.2) has at least one solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M21">View MathML</a>.

Superlinear problems with classical boundary value conditions have been considered in many papers, particularly in the second and fourth order cases, with either periodic or separated boundary conditions, see for example [2-11] and the references therein. Specifically, the second order periodic problem is considered in [2,3], while [4-7] consider problems with separated boundary conditions, and results similar to Theorem 1.1 were obtained in each of these papers. The fourth order periodic problem is considered in [8-10]. Rynne [11] and De Coster [12] consider some general higher order problems with separated boundary conditions also.

Calvert and Gupta [13] studied the superlinear three-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M22">View MathML</a>

(1.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M23">View MathML</a>

(1.8)

(which is a nonlocal boundary value problem), under the assumptions:

(A0) β ∈ (0, 1) ∪ (1, ∞);

(A1) g : ℝ → ℝ is continuous and satisfies g(s)s > 0, s ≠ 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M24">View MathML</a>is increasing and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M25">View MathML</a>

(A2) p : [0, 1] × ℝ2 → ℝ is a function satisfying the Carathéodory conditions and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M26">View MathML</a>

where M1 : [0, 1] × [0, ∞) → [0, ∞) satisfies the condition: for each s ∈ [0, ∞), M1(·, s) is integrable on [0, 1] and for each t ∈ [0, 1], M1(t, ·) is increasing on [0, ∞) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M27">View MathML</a> as s → ∞.

Calvert and Gupta used Leray-Schauder degree and some ideas from Henrard [14] and Cappieto et al. [5] to prove the existence of infinity many solutions for (1.7), (1.8). Their results extend the main results in [14].

It is the purpose of this paper to use the global bifurcation theorem, see [15] and [1], to obtain infinity many nodal solutions to m-point boundary value problems (1.1), (1.2) under the assumptions (H1)-(H3). Obviously, our conditions (H2) and (H3) are much weaker than the corresponding restrictions imposed in [13]. Our paper uses some of ideas of Rynne [10], which deals with fourth order two-point boundary value problems. By the way, the proof [10, Lemma 2.8] contains a small error (since ||u″|0 ζ4(0) ⇏ |u″|0 ζ4(R) there). So, we introduce a new function χ (see (3.7)) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M28">View MathML</a>

which are required in applying Lemma 3.4.

2 Eigenvalues of the linear problem

First, we state some preliminary results related to the linear eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M29">View MathML</a>

(2.1)

Denote the spectrum of L by σ(L). The following spectrum results on (2.1) were established by Rynne [1], which extend the main result of Ma and O'Regan [16].

Lemma 2.1. [1, Theorem 3.1] The spectrum σ(L) consists of a strictly increasing sequence of eigenvalues λk > 0, k = 1, 2, ..., with corresponding eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M30">View MathML</a>. In addition,

(i) limk→∞ λk = ∞;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M31">View MathML</a>, for each k ≥ 1, and ϕ1 is strictly positive on (0, 1).

Lemma 2.2 [1, Theorem 3.8] For each k ≥ 1, the algebraic multiplicity of the characteristic value λk of L-1 : Y Y is equal to 1.

3 Proof of the main results

For any u X, we define e(u)(·): [0, 1] → ℝ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M32">View MathML</a>

It follows from (1.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M33">View MathML</a>

(3.1)

For any s ∈ ℝ, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M34">View MathML</a>

and for any s ≥ 0, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M35">View MathML</a>

We now consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M36">View MathML</a>

(3.2)

where α ∈ [0, 1] is an arbitrary fixed number and λ ∈ ℝ. In the following lemma (λ, u) ∈ ℝ × X will be an arbitrary solution of (3.2).

By (H2), we can choose b1 ≥ 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M37">View MathML</a>

(3.3)

By (1.2), we have the following

Lemma 3.1. Let (H1) hold and let u X. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M38">View MathML</a>

(3.4)

Lemma 3.2. Let u be a solution of (3.2). Then for any x0, x1 ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M39">View MathML</a>

Proof. Multiply (3.2) by u' and integrate from x0 to x1, then we get the desired result. ■

In the following, let us fix R ∈ (0, ∞) so large that R b1 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M40">View MathML</a>

(3.5)

Lemma 3.3. There exists an increasing function ζ1 : [0, ∞) → [0, ∞), such that for any solution u of (3.2) with 0 ≤ λ R and |u(x0)| + |u'(x0)| ≤ R for some x0 ∈ [0, 1], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M41">View MathML</a>

Proof. Choose x1 ∈ [0, 1] such that |u'|0 = |u'(x1)|. We obtain from Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M42">View MathML</a>

Combining this with (3.1), (3.4), it concludes that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M43">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M44">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M45">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M46">View MathML</a>

(3.6)

Clearly, the function is nondecreasing.

Lemma 3.4 Let u be a solution of (3.2) with 0 ≤ λ R and |u'|0 ζ2(R) for some R > 0. Then, for any x ∈ [0, 1] with |u(x)| ≤ R, we have |u'(x)| ≥ R2.

Proof. Suppose, on the contrary that there exists x0 ∈ (0, 1) such that |u(x0)| ≤ R and |u'(x0)| < R2. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M47">View MathML</a>

Combining this with λ R < R + R2 and using Lemma 3.3, it concludes that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M48">View MathML</a>

However, this is impossible if |u'|0 ζ2(R). ■

For fixed R > b1, let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M49">View MathML</a>

(3.7)

Let us now consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M50">View MathML</a>

(3.8)

where θ : ℝ → ℝ is a strictly increasing, C-function with θ(s) = 0, s ≤ 1 and θ(s) = 1, s ≥ 2. The nonlinear term in (3.8) is a continuous function of (λ, u) ∈ ℝ × X and is zero for λ ∈ ℝ, |u'|0 χ(λ), so (3.8) becomes a linear eigenvalue problem in this region, and overall the problem can be regarded as a bifurcation (from u = 0) problem.

The next lemma now follows immediately.

Lemma 3.5 The set of solutions (λ, u) of (3.8) with |u'|0 χ(λ) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M51">View MathML</a>

We also have the following global bifurcation result for (3.8).

Lemma 3.6 For each k ≥ 1 and ν ∈ {+, -}, there exists a connected set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M52">View MathML</a> of nontrivial solutions of (3.8) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M53">View MathML</a> is closed and connected and:

(i) there exists a neighborhood Nk of (λk, 0) in ℝ × E such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M54">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> meets infinity in ℝ × E (that is, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M56">View MathML</a>, such that |λn| + |un|E → ∞).

Proof. Since L-1 : Y X exists and is bounded, (3.8) can be rewritten in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M57">View MathML</a>

(3.9)

and since L-1 can be regarded as a compact operator from Y to E, it is clear that finding a solution (λ, u) of (3.8) in ℝ × E is equivalent to finding a solution of (3.9) in ℝ × E. Now, by the similar method used in the proof of [1, Theorem 4.2]), we may deduce the desired result.

Since e(u)(t) σ 0 in (3.8), nodal properties need not be preserved. However, we will rely on preservation of nodal properties for "large" solutions, encapsulated in the following result.

Lemma 3.7 If (λ, u) is a solution of (3.8) with λ ≥ 0 and |u'|0 > χ(λ), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58">View MathML</a>, for some k ≥ 1 and ν ∈ {+, -}.

Proof. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M59">View MathML</a> for any k ≥ 1 and ν, then one of the following cases must occur:

Case 1. u'(0) = 0;

Case 2. u' (τ) = u″(τ) = 0 for some τ ∈ (0, 1].

In the Case 1, u(t) ≡ 0 on [0, 1]. This contradicts the assumption |u'|0 > χ(λ) ≥ ζ2(λ). So this case cannot occur.

In the Case 2, we have from (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M60">View MathML</a>

(3.10)

Since |u'|0 > χ(λ), we have from the definition of θ that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M61">View MathML</a>

(3.11)

It follows from Lemma 3.4 that |u(τ)| > R b1. Combining this with (3.11) and (3.3), it concludes that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M62">View MathML</a>

(3.12)

which contradicts (3.10). So, Case 2 cannot occur.

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58">View MathML</a> for any k ≥ 1 and ν ∈ {+, -}. ■

In view of Lemmas 3.5 and 3.7, in the following lemma, we suppose that (λ, u) is an arbitrary nontrivial solution of (3.8) with λ ≥ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58">View MathML</a>, for some k ≥ 1 and ν.

Lemma 3.8. There exists an integer k0 ≥ 1 (depending only on χ(0)) such that for any nontrivial solution u of (3.8) with λ = 0 and χ(0) ≤ |u'|0 ≤ 2χ(0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M63">View MathML</a>

(3.13)

Proof. Let x1, x2 be consecutive zeros of u. Then there exists x3 ∈ (x1, x2) such that u'(x3) = 0, and hence, Lemma 3.4, (3.3), and (3.7) yield that |u(x3)| > 1. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M64">View MathML</a>

for some τ1 ∈ (x3, x2), τ2 ∈ (x1, x3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M65">View MathML</a>

(3.14)

Notice that |u'|0 > χ(0) ≥ ζ2(R) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M58">View MathML</a> for some k ∈ ∞ and ν ∈ {+, -}, and subsequently, there exist 0 < r1 < r2 < · · · < rk-1, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M66">View MathML</a>

This together with (3.14) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M67">View MathML</a>

and accordingly, k < |u'|0/2 + 1 ≤ χ(0) + 1. ■

Now let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M68">View MathML</a>

Lemma 3.9. Suppose that 0 ≤ λ R and |u'|0 χ(R). Then WR(u) consists of at least k intervals and at most k + 1 intervals, each of length less than 2/R, and VR(u) consists of at least k intervals and at most k + 1 intervals.

Proof. Lemma 3.4 implies that |u'(x)| ≥ R2 for all x WR(u). For any interval I WR(u), u' does not change sign on I, say,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M69">View MathML</a>

We claim that the length of I is less than 2/R.

In fact, for x, y I with x > y, say,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M70">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M71">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M72">View MathML</a>

The case

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M73">View MathML</a>

can be treated by the similar method. Since u is monotonic in any subinterval containing in WR(u), the desired result is followed. ■

Lemma 3.10. There exists ζ3 with limR→∞ ζ3(R) = 0, and η1 ≥ 0 such that, for any R η1, if either

(a) 0 ≤ λ R and |u'|0 = 2χ(R), or

(b) λ = R and χ(R) ≤ |u'|0 ≤ 2χ(R),

then the length of each interval of VR(u) is less than ζ3(R).

Proof. Define H = H(R) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M74">View MathML</a>

and let ζ3(R) := 2π/H(R). By (1.4), limR→∞ H(R) = ∞, so limR→∞ζ3(R) = 0, and we may choose η1 b1 sufficiently large that H(R) > 0 for all R η1.

We firstly show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M75">View MathML</a>

(3.15)

In fact, if |u(x)| ≤ R on [0, 1], then Lemma 3.4 yields that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M76">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M77">View MathML</a>

However, these contradict the boundary conditions (1.2), since (H1) implies u'(s0) = 0 for some s0 ∈ (0, 1). Therefore, (3.15) is valid.

Now, Let us choose x0, x2 such that either

(1) u(x0) = u(x2) = R and u > R on (x0, x2) or

(2) u(x0) = R, x2 = 1 and u > R on (x0, 1].

(the case of intervals on which u < 0 is similar). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M78">View MathML</a>

By (3.8) and the construction of H(R), if either (a) or (b) holds then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M79">View MathML</a>

and by Lemma 3.4, u'(x0) > 0, and u'(x2) < 0, if x2 < 1.

Suppose, now on the contrary that x2 - x0 > ζ3(R), that is, l := 2π/(x2 - x0) < H(R). Defining x1 = (x0 + x2)/2 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M80">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M81">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M82">View MathML</a>

and this contradiction shows that x2 - x0 ζ3(R), which proves the lemma.

Now, we are in the position to prove Theorem 1.1.

Proof of Theorem 1.1 Now, choose an arbitrary integer k k0 and ν ∈ {+, -}, and choose Λ > max{η1, μk} (Here, we assume Λ > η1, so that Lemma 3.10 could be applied!) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M83">View MathML</a>

(3.16)

Notice that Lemma 3.9 implies that if |u'|0 χ(Λ), then the length of each interval of WΛ(u) is less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M84">View MathML</a> for 0 ≤ λ ≤ Λ. This together with (3.16) and Lemma 3.10 imply that there exists no solution (λ, u) of (3.8), which satisfies either

(a) 0 ≤ λ ≤ Λ and |u'|0 = 2χ(Λ) or

(b) λ = Λ and χ(Λ) ≤ |u'|0 ≤ 2χ(Λ).

Now, let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M85">View MathML</a>

It follows from Lemma 3.5 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> "enters" B through the set D1, while from Lemma 3.7, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M86">View MathML</a>. Thus, by Lemma 3.6 and the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M87">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> must "leave" B. (Suppose, on the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> does not "leave" B, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M88">View MathML</a>

which contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> joins (μk, 0) to infinity in ℝ × E.) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> is connected, it must intersect ∂B. However, Lemmas 3.8-3.10 (together with (3.16)) show that the only portion of ∂B (other than D1), which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M55">View MathML</a> can intersect is D2. Thus, there exists a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M89">View MathML</a>, and clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/14/mathml/M90">View MathML</a> provides the desired solution of (1.1)-(1.2).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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

Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. Supported by the NSFC(No.11061030), the Fundamental Research Funds for the Gansu Universities.

References

  1. Rynne, BP: Spectral properties and nodal solutions for second-order, m-point, boundary value problems. Nonlinear Anal. Theory Method Appl. 67(12), 3318–3327 (2007). Publisher Full Text OpenURL

  2. Cappieto, A, Mawhin, J, Zanolin, F: A continuation approach to superlinear periodic boundary value problems. J Diff Equ. 88, 347–395 (1990). Publisher Full Text OpenURL

  3. Ding, T, Zanolin, F: Periodic solutions of Duffing's equation with superquadratic potential. J. Diff. Equ. 97, 328–378 (1992). Publisher Full Text OpenURL

  4. Pimbley, GH Jr.: A superlinear Sturm-Liouville problem. Trans. Am. Math. Soc. 103, 229–248 (1962). Publisher Full Text OpenURL

  5. Cappieto, A, Henrard, M, Mawhin, J, Zanolin, F: A continuation approach to some forced superlinear Sturm-Liouville boundary value problems. Topol. Methods Nonlinear Anal. 3, 81–100 (1994)

  6. Cappieto, A, Mawhin, J, Zanolin, F: On the existence of two solutions with a prescribed number of zeros for a superlinear two point boundary value problem. Topol. Methods Nonlinear Anal. 6, 175–188 (1995)

  7. Ma, R, Thompson, B: Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities. J. Math. Anal. Appl. 303(2), 726–735 (2005). Publisher Full Text OpenURL

  8. Mawhin, J, Zanolin, F: A continuation approach to fourth order superlinear periodic boundary value problems. Topol. Methods Nonlinear Anal. 2, 55–74 (1993)

  9. Conti, M, Terracini, S, Verzini, G: Infinitely many solutions to fourth order superlinear periodic problems. Trans. Am. Math. Soc. 356, 3283–3300 (2004). Publisher Full Text OpenURL

  10. Rynne, BP: Infinitely many solutions of superlinear fourth order boundary value problems. Topol. Methods Nonlinear Anal. 19(2), 303–312 (2002)

  11. Rynne, BP: Global bifurcation for 2mth order boundary value problems and infinitely many solutions of superlinear problems. J. Diff. Equ. 188(2), 461–472 (2003). Publisher Full Text OpenURL

  12. De Coster, C, Gaudenzi, M: On the existence of infinitely many solutions for superlinear nth order boundary value problems. Nonlinear World. 4, 505–524 (1997)

  13. Calverta, B, Gupta, CP: Multiple solutions for a super-linear three-point boundary value problem. Nonlinear Anal. Theory Method Appl. 50, 115–128 (2002). Publisher Full Text OpenURL

  14. Henrard, M: Topological degree in boundary value problems: existence and multiplicity results for second-order differential equations. In: Henrard ME (ed.) Thesis,p. 123. Université Catholique de Louvain, Belgium (1995)

  15. Rabinowitz, PH: Some global results for nonlinear eigenvalue problems. J Funct Anal. 7(3), 487–513 (1971). Publisher Full Text OpenURL

  16. Ma, R, O'Regan, D: Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues. Nonlinear Anal Theory Method Appl. 64, 1562–1577 (2006). Publisher Full Text OpenURL