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New results on anti-periodic boundary value problems for second-order nonlinear differential equations

Ruixi Liang

Author affiliations

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410075, China

Citation and License

Boundary Value Problems 2012, 2012:112  doi:10.1186/1687-2770-2012-112

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/112


Received:27 March 2012
Accepted:27 September 2012
Published:11 October 2012

© 2012 Liang; 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

This paper is concerned with the solvability of anti-periodic boundary value problems for second-order nonlinear differential equations. By using topological methods, some sufficient conditions for the existence of solution are obtained, which extend and improve the previous results.

MSC: 34B05.

Keywords:
anti-periodic boundary value problem; existence of solution; nonlinear

1 Introduction

In this paper, we will consider the existence of solutions to second-order differential equations of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M1">View MathML</a>

(1.1)

subject to the anti-periodic boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M2">View MathML</a>

(1.2)

where T is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M3">View MathML</a> is continuous. Equation (1.1) subject to (1.2) is called an anti-periodic boundary value problem.

Anti-periodic problems have been studied extensively in recent years. For example, anti-periodic boundary value problems for ordinary differential equations were considered in [1-9]. Also, anti-periodic boundary conditions for impulsive differential equations, partial differential equations and abstract differential equations were investigated in [10-16]. The methods and techniques employed in these papers involve the use of the Leray-Schauder degree theory [7,8], the upper and lower solutions [2,10-12], and a fixed point theorem [1]. By using Schauder’s fixed point theorem and lower and upper solutions method, Wang and Shen in [3] considered the anti-periodic boundary value problem (1.1) and (1.2) when a first-order derivative is not involved explicitly in the nonlinear term f, namely equation (1.1) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M4">View MathML</a>

(1.3)

They proved the following theorems.

Theorem 1.1 ([[3], Theorem 2.2])

Assume there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M6">View MathML</a>, and functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M7">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M8">View MathML</a>

(1.4)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M9">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M10">View MathML</a>. Further suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M11">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M12">View MathML</a>. Then (1.2) and (1.3) have at least one solution.

Theorem 1.2 ([[3], Theorem 1.2])

Letγbe a positive constant. Assume there exist a continuous and nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M13">View MathML</a>and a nonnegative function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M14">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M15">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M9">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M17">View MathML</a>. Further suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M18">View MathML</a>

(1.5)

Then (1.2) and (1.3) have at least one solution.

In this paper, we are interested in the existence of a solution to the anti-periodic boundary value problem (1.1) and (1.2). The significant point here is that the right-hand side of (1.1) may depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19">View MathML</a>. The dependence of right-hand side on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19">View MathML</a> is naturally seen in many physical phenomena, and we refer the readers to [17,18] for some nice examples. If there appears <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M19">View MathML</a> in nonlinear term, the relative boundary value problem will be more complicated. Meanwhile, we note equation (1.4) or (1.5) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M22">View MathML</a> is at most linear for x, so the problem has not been solved when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M22">View MathML</a> is super-linear for x. Motivated by the above two aspects, we devote ourselves to studying the anti-periodic boundary value problem (1.1) and (1.2).

The paper is organized as follows. In Section 2, we reformulate the anti-periodic boundary value problem (1.1) and (1.2) as an equivalent integral equation, which is a widely used technique in the theory of boundary value problem. In Section 3, a general existence result is presented for (1.1) and (1.2). The result provides a natural motivation for the obtention of a priori bounds on solutions and greatly minimizes the proofs of the new results in the following section. The main tool used here is the Leray-Schauder topological degree. In Section 4, some new conditions are presented for (1.1) and (1.2). The new conditions involve linear or quadratic growth constraints on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M24">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M25">View MathML</a>.

2 Preliminaries

If a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M26">View MathML</a> satisfies equations (1.1) and (1.2), we call x a solution of (1.1) and (1.2). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M27">View MathML</a> be a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M28">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M30">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M32">View MathML</a> and consider the anti-periodic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M33">View MathML</a>

(2.1)

Lemma 2.1xis a solution of (2.1) if and only ifxsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M34">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M35">View MathML</a>

Proof Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> is a solution of (2.1) and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M37">View MathML</a>, then the first equation of (2.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M38">View MathML</a>

(2.3)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M39">View MathML</a>

(2.4)

then from (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M40">View MathML</a>

Multiplying both sides of the above equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M41">View MathML</a> and integrating from 0 to t yields

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M43">View MathML</a>.

Similarly, multiplying the two sides of (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M44">View MathML</a> and integrating from 0 to t yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M45">View MathML</a>

(2.5)

By direct computation, we get

(2.6)

Substituting (2.6) into (2.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M47">View MathML</a>

(2.7)

Hence,

Further from (2.7),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M49">View MathML</a>

and therefore

Taking into account <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M52">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M53">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M54">View MathML</a>

(2.9)

Substituting (2.8) and (2.9) into (2.7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M55">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> is a solution of (2.2).

On the other hand, assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> is a solution of (2.2). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M58">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M59">View MathML</a>

And

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M60">View MathML</a>

Direct computation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M61">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> is a solution of (2.1). This proof is complete. □

For later use, we present the following estimations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M63">View MathML</a>

(2.10)

Remark 2.1 The integral equation (2.2) we obtained is much simpler than that in [3] which needs a double integral.

Combining Lemma 2.1 and equation (1.1), we can easily get

Theorem 2.1The anti-periodic boundary value problem (1.1) and (1.2) is equivalent to the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M64">View MathML</a>

(2.11)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M66">View MathML</a>is defined in Lemma 2.1.

Define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M67">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M68">View MathML</a>

(2.12)

Lemma 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M69">View MathML</a>is completely continuous.

Proof Noting the continuity of f, this follows in a standard step-by-step process and so is omitted. □

In view of Theorem 2.1, we obtain

Theorem 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M26">View MathML</a>is a solution of the anti-periodic boundary value problem (1.1) and (1.2) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M71">View MathML</a>is the fixed point of the operatorT.

3 General existence

In this section, an abstract existence result is presented for (1.1) and (1.2). The obtained result emphasizes the natural search for a priori bounds on solutions to the boundary value problem, which will be conducted in the following section.

Firstly, we introduce some basic properties of the Leray-Schauder degree. For more detail, we refer an interested reader to [19,20].

Theorem 3.1The Leray-Schauder degree has the following properties.

(i) (Homotopy invariance) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M72">View MathML</a>be a bounded open set, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M73">View MathML</a>be compact. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M74">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M75">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M76">View MathML</a>is independent oft.

(ii) (Existence) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M77">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M78">View MathML</a>.

Now, we give the main result of this section.

Theorem 3.2LetM, Nandλbe positive constants inRand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M79">View MathML</a>be continuous. Consider the family of anti-periodic boundary value problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M80">View MathML</a>

(3.1)

If all potential solutions to (3.1) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M81">View MathML</a>

withMandNindependent ofμ, then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.

Proof In view of Theorem 2.2, we want to show there exists at least one <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M71">View MathML</a> with x satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M83">View MathML</a>. This solution will then naturally be in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M84">View MathML</a>.

Consider the family of problems associated with (2.12), namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M85">View MathML</a>

(3.2)

Note that (3.2) is equivalent to the family of anti-periodic boundary value problems (3.1).

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M86">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M87">View MathML</a>

From Lemma 2.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M88">View MathML</a> is completely continuous. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M89">View MathML</a> is a compact mapping. By the assumption of the theorem, all possible solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M90">View MathML</a> must satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M91">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M92">View MathML</a>

Hence, the following Leray-Schauder degrees are defined and the homotopy invariance principle in Theorem 3.1 applies:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M93">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M94">View MathML</a>. By the existence property of the Leray-Schauder degree, (3.2) has at least one solution in Ω for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M95">View MathML</a>. And hence (1.1) and (1.2) has at least one solution. □

4 Main results

In this section, some existence theorems are presented.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M98">View MathML</a>be nonnegative constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31">View MathML</a>. Iffis continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M100">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M102">View MathML</a>

then the anti-periodic boundary problem (1.1) and (1.2) has at least one solution.

Proof Consider the family (3.1). We want to show the conditions of Theorem 3.2 hold for some positive constants M and N.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> be a solution to (3.1) and consider the equivalent equation (3.2), that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M104">View MathML</a>

(4.1)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M105">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M106">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M107">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M108">View MathML</a>

(4.2)

Differentiating both sides of (4.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M109">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M110">View MathML</a>

and because of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M111">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M112">View MathML</a>

The rearrangement yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M113">View MathML</a>

(4.3)

By substituting (4.3) into (4.2) and rearranging, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M114">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M115">View MathML</a>

Hence, Theorem 3.2 holds for positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M117">View MathML</a>. The solvability of (1.1) and (1.2) now follows. □

Theorem 4.2Assume there exist nonnegative constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M119">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M121">View MathML</a>

then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.

Proof Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M36">View MathML</a> is a solution of (3.1), and in view of (2.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M123">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M124">View MathML</a>

Therefore, Theorem 3.2 holds for positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M126">View MathML</a>. The solvability of (1.1) and (1.2) now follows. □

Example 4.1 Consider the anti-periodic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M127">View MathML</a>

(4.4)

We claim (4.4) has at least one solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M128">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M129">View MathML</a> in Theorem 4.2. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M130">View MathML</a>, we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M131">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M132">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M133">View MathML</a>

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M134">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M135">View MathML</a>. Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M137">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M138">View MathML</a>

Then the conclusion follows from Theorem 4.2. □

Now, we reconsider the problem (1.2) and (1.3). The following result is obtained.

Theorem 4.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M139">View MathML</a>is continuous. If there exist nonnegative constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M141">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M31">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M143">View MathML</a>

then (1.2) and (1.3) has at least one solution.

Proof The proof is similar to Theorem 4.2 and here we omit it. □

An example to highlight the Theorem 4.3 is presented.

Example 4.2 Consider the anti-periodic boundary value problem given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M144">View MathML</a>

(4.5)

We claim (4.5) has at least one solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M145">View MathML</a> and see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M146">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M147">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M141">View MathML</a> and λ to be chosen below, see that

Thus, the conditions of Theorem 4.3 hold and the solvability follows. □

Remark 4.1 The results of [3] do not apply to the above example since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M151">View MathML</a> grows more than linearly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M152">View MathML</a>. Therefore, we improve the previous results.

Finally, in order to illustrate our main results, we use the ‘bvp4c’ package in MATLAB to simulate. As shown in Figure 1(a) and (b), numerical simulations also suggest that Examples 4.1 and 4.2 with the given coefficients admit at least one solution.

thumbnailFigure 1. Solutions found by numerical stimulations with (a):<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M153">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M154">View MathML</a>in equation (4.5); (b):<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M155">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/112/mathml/M156">View MathML</a>in equation (4.4).

Competing interests

The author declares that they have no competing interests.

Author’s contributions

The author typed, read and approved the final manuscript.

Acknowledgements

The author would like to thank anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of work. This research was partially supported by the NNSF of China (No. 11001274) and the Postdoctoral Science Foundation of Central South University and China (No. 2011M501280).

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