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Existence of fast homoclinic orbits for a class of second-order non-autonomous problems

Qiongfen Zhang1*, Qi-Ming Zhang2 and Xianhua Tang3

Author Affiliations

1 College of Science, Guilin University of Technology, Guilin, Guangxi, 541004, P.R. China

2 College of Science, Hunan University of Technology, Zhuzhou, Hunan, 412000, P.R. China

3 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China

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Boundary Value Problems 2014, 2014:89  doi:10.1186/1687-2770-2014-89

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/89


Received:3 January 2014
Accepted:23 April 2014
Published:6 May 2014

© 2014 Zhang 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 credited.

Abstract

By applying the mountain pass theorem and the symmetric mountain pass theorem in critical point theory, the existence and multiplicity of fast homoclinic solutions are obtained for the following second-order non-autonomous problem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6">View MathML</a> are not periodic in t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7">View MathML</a> is a continuous function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M8">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M9">View MathML</a>.

MSC: 34C37, 35A15, 37J45, 47J30.

Keywords:
fast homoclinic solutions; variational methods; critical point

1 Introduction

Consider fast homoclinic solutions of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M10">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M6">View MathML</a> are not periodic in t, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M7">View MathML</a> is a continuous function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M17">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M18">View MathML</a>

(1.2)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M19">View MathML</a>, problem (1.1) reduces to the following special second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M20">View MathML</a>

(1.3)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M21">View MathML</a>, problem (1.1) reduces to the following second-order damped vibration problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M22">View MathML</a>

(1.4)

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M19">View MathML</a>, then problem (1.1) reduces to the following second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M25">View MathML</a>

(1.5)

The existence of homoclinic orbits plays an important role in the study of the behavior of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic. If it has smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. The first work about homoclinic orbits was done by Poincaré [1].

Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [2-30] and references therein. In [6,16,17], the authors considered homoclinic solutions for the special Hamiltonian system (1.3) in weighted Sobolev space. Later, Shi et al.[31] obtained some results for system (1.3) with a p-Laplacian, which improved and generalized the results in [6,16,17]. However, there is little research as regards the existence of homoclinic solutions for damped vibration problems (1.4) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M26">View MathML</a>. In 2008, Wu and Zhou [32] obtained some results for damped vibration problems (1.4) with some boundary value conditions by variational methods. Zhang and Yuan [33,34] studied the existence of homoclinic solutions for (1.4) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M27">View MathML</a> is a constant. Later, Chen et al.[35] investigated fast homoclinic solutions for (1.4) and obtained some new results under more relaxed assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M28">View MathML</a>, which resolved some open problems in [33]. Zhang [36] obtained infinitely many solutions for a class of general second-order damped vibration systems by using the variational methods. Zhang [37] investigated subharmonic solutions for a class of second-order impulsive systems with damped term by using the mountain pass theorem.

Motivated by [21,23,32-34,38-42], we will establish some new results for (1.1) in weighted Sobolev space. In order to introduce the concept of fast homoclinic solutions for problem (1.1), we first state some properties of the weighted Sobolev space E on which the certain variational functional associated with (1.1) is defined and the fast homoclinic solutions are the critical points of the certain functional.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M29">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M30">View MathML</a> is defined in (1.2) and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M31">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M32">View MathML</a>

Then X is a Hilbert space with the norm given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M33">View MathML</a>

It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M34">View MathML</a>

with the embedding being continuous. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M35">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M36">View MathML</a>) denotes the Banach spaces of functions on ℝ with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M37">View MathML</a> under the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M38">View MathML</a>

If σ is a positive, continuous function on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M39">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M40">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M41">View MathML</a> equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M42">View MathML</a>

is a reflexive Banach space.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M43">View MathML</a>, where a is the function given in condition (A). Then E with its standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M44">View MathML</a> is a reflexive Banach space. Similar to [33,35], we have the following definition of fast homoclinic solutions.

Definition 1.1 If (1.2) holds, a solution of (1.1) is called a fast homoclinic solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a>.

The functional φ corresponding to (1.1) on E is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M46">View MathML</a>

(1.6)

Clearly, it follows from (W1) or (W1)′ that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M47">View MathML</a>. By Theorem 2.1 of [43], we can deduce that the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M48">View MathML</a>

is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M49">View MathML</a> in the dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M50">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M51">View MathML</a>. As the embeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M52">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M53">View MathML</a> are continuous, if (A) and (W1) or (W1)′ hold, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M54">View MathML</a> and one can easily check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M55">View MathML</a>

(1.7)

Furthermore, the critical points of φ in E are classical solutions of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M56">View MathML</a>.

Now, we state our main results.

Theorem 1.1Suppose thata, q, andWsatisfy (1.2) and the following conditions:

(A) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M58">View MathML</a>is a continuous, positive function onsuch that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M60">View MathML</a>

(W1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M62">View MathML</a>, and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M63">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M64">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M65">View MathML</a>.

(W2) There is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M67">View MathML</a>

(W3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M68">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M69">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M70">View MathML</a>

Then problem (1.1) has at least one nontrivial fast homoclinic solution.

Theorem 1.2Suppose thata, q, andWsatisfy (1.2), (A), (W2), and the following conditions:

(W1)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M62">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M73">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>.

(W3)′ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M68">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M69">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M77">View MathML</a>

Then problem (1.1) has at least one nontrivial fast homoclinic solution.

Theorem 1.3Suppose thata, q, andWsatisfy (1.2), (A), (W1)-(W3), and the following assumption:

(W4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M79">View MathML</a>.

Then problem (1.1) has an unbounded sequence of fast homoclinic solutions.

Theorem 1.4Suppose thata, q, andWsatisfy (1.2), (A), (W1)′, (W2), (W3)′, and (W4). Then problem (1.1) has an unbounded sequence of fast homoclinic solutions.

Remark 1.1 It is easy to see that our results hold true even if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M23">View MathML</a>. To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature, from this point, our results are new. As pointed out in [17], condition (A) can be replaced by more general assumption: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M81">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M82">View MathML</a>.

The rest of this paper is organized as follows: in Section 2, some preliminaries are presented. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

2 Preliminaries

Let E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M44">View MathML</a> be given in Section 1, by a similar argument in [41], we have the following important lemma.

Lemma 2.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M85">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M86">View MathML</a>

(2.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M87">View MathML</a>

(2.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M89">View MathML</a>.

The following lemma is an improvement result of [16].

Lemma 2.2Ifasatisfies assumption (A), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M90">View MathML</a>

(2.4)

Moreover, there exists a Hilbert spaceZsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M91">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M92">View MathML</a>

(2.6)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M94">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M95">View MathML</a>

where from (A) and (1.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M96">View MathML</a>. Then (2.4) holds.

By (A), there exists a continuous positive function ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M97">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M82">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M99">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M100">View MathML</a>

(2.5) holds by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M101">View MathML</a>.

Finally, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M102">View MathML</a> is the weighted Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M103">View MathML</a>, it follows from [43] that (2.6) holds. □

The following two lemmas are the mountain pass theorem and the symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3[44]

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M104">View MathML</a>satisfying (PS)-condition. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M105">View MathML</a>and:

(i) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M107">View MathML</a>.

(ii) There exists an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M108">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M109">View MathML</a>.

ThenIpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M110">View MathML</a>which can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M111">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M113">View MathML</a>is an open ball inEof radiusρcentered at 0.

Lemma 2.4[44]

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M104">View MathML</a>withIeven. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M105">View MathML</a>andIsatisfies (PS)-condition, assumption (i) of Lemma 2.3 and the following condition:

(iii) For each finite dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M116">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M117">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M118">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M120">View MathML</a>is an open ball inEof radiusrcentered at 0.

ThenIpossesses an unbounded sequence of critical values.

Lemma 2.5Assume that (W2) and (W3) or (W3)′ hold. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M121">View MathML</a>,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M122">View MathML</a>is nondecreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M123">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M124">View MathML</a>is nonincreasing on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M123">View MathML</a>.

The proof of Lemma 2.5 is routine and we omit it.

3 Proofs of theorems

Proof of Theorem 1.1 Step 1. The functional φ satisfies the (PS)-condition. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M126">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M127">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M128">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M129">View MathML</a>. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M131">View MathML</a>

(3.1)

From (1.6), (1.7), (3.1), (W2), and (W3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M132">View MathML</a>

It follows from Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M133">View MathML</a>, and the above inequalities that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M134">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M135">View MathML</a>

(3.2)

Now we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136">View MathML</a> in E. Passing to a subsequence if necessary, it can be assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M137">View MathML</a> in E. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M138">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M139">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M140">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M141">View MathML</a>

(3.3)

It follows from (2.2), (3.2), and (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M142">View MathML</a>

(3.4)

Similarly, by (2.3), (3.2), and (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M143">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M137">View MathML</a> in E, it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M145">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M146">View MathML</a> pointwise for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M147">View MathML</a>. Hence, it follows from (3.4) and (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M148">View MathML</a>

(3.6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M149">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M150">View MathML</a>, the operator defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M151">View MathML</a> is a linear continuous map. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M153">View MathML</a>. The Sobolev theorem implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136">View MathML</a> uniformly on J, so there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M155">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M156">View MathML</a>

(3.7)

For any given number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M157">View MathML</a>, by (W1), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M158">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M159">View MathML</a>

(3.8)

From (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M160">View MathML</a>

(3.9)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M58">View MathML</a> is a positive continuous function on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M145">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M146">View MathML</a> pointwise for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M164">View MathML</a>, it follows from (3.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M165">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M166">View MathML</a>

From Lebesgue’s dominated convergence theorem, (3.4), (3.5), (3.6), (3.9), and the above inequalities, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M167">View MathML</a>

(3.10)

From Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M169">View MathML</a>. Hence, by (3.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M170">View MathML</a>

tends to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M171">View MathML</a>, which together with (3.7) shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M172">View MathML</a>

(3.11)

From (1.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M173">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M174">View MathML</a> is a positive constant. It follows from (3.11) and (3.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M175">View MathML</a>

(3.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M176">View MathML</a>

(3.14)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M136">View MathML</a> in E by (3.13) and (3.14). This shows that φ satisfies (PS)-condition.

Step 2. From (W1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M179">View MathML</a>

(3.15)

By (3.15) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M181">View MathML</a>

(3.16)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M182">View MathML</a>

(3.17)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M184">View MathML</a>, it follows from Lemma 2.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M185">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>. From Lemma 2.5(i) and (3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M187">View MathML</a>

(3.18)

By (W3), (3.16), and (3.18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M188">View MathML</a>

Therefore, we can choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M189">View MathML</a> depending on ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M190">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M192">View MathML</a>.

Step 3. From Lemma 2.5(ii) and (2.1), we have for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M194">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M196">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M197">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M198">View MathML</a>

(3.20)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M200">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M201">View MathML</a>, from Lemma 2.5(i) and (3.20), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M202">View MathML</a>

(3.21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M203">View MathML</a>. From (W3), (1.6), (3.19), (3.20), and (3.21), we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M201">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M205">View MathML</a>

(3.22)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M207">View MathML</a>, it follows from (3.22) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M208">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M209">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M210">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M211">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M213">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M214">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M215">View MathML</a>. By Lemma 2.3, φ has a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M216">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M217">View MathML</a>

(3.23)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M218">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M219">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M220">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M221">View MathML</a> is the desired solution of problem (1.1). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M221">View MathML</a> is a nontrivial fast homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M224">View MathML</a> in (W3) is only used in the proofs of (3.2) and Step 2. Therefore, we only need to prove that (3.2) and Step 2 still hold if we use (W1)′ and (W3)′ instead of (W1) and (W3). We first prove that (3.2) holds. From (W2), (W3)′, (1.6), (1.7), and (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M225">View MathML</a>

which implies that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M226">View MathML</a> such that (3.2) holds. Next, we prove that Step 2 still holds. From (W1)′, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M228">View MathML</a>

(3.24)

By (3.24) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M180">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M230">View MathML</a>

(3.25)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M231">View MathML</a>, it follows from Lemma 2.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M232">View MathML</a>. It follows from (1.6) and (3.25) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M233">View MathML</a>

Therefore, we can choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M189">View MathML</a> depending on ρ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M190">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M237">View MathML</a>. The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M238">View MathML</a> and satisfies (PS)-condition and assumption (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239">View MathML</a> be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M240">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M241">View MathML</a>

(3.26)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M243">View MathML</a> is a basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M245">View MathML</a>

(3.27)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M246">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M248">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M249">View MathML</a>

(3.28)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M250">View MathML</a>

(3.29)

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M251">View MathML</a> is a norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M239">View MathML</a>. Hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M253">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M254">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M255">View MathML</a>, by Lemma 2.1, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M256">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M257">View MathML</a>

(3.30)

where δ is given in (3.25). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M258">View MathML</a>

(3.31)

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M260">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M261">View MathML</a>

(3.32)

Then by (3.26)-(3.29), (3.31), and (3.32), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M262">View MathML</a>

(3.33)

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M263">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M264">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M265">View MathML</a>, which together with (3.30), implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M266">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M267">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M268">View MathML</a>

(3.34)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M269">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M271">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M272">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M273">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M45">View MathML</a>, from Lemma 2.1 and Lemma 2.5(i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M275">View MathML</a>

(3.35)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M277">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M248">View MathML</a>, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M280">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M281">View MathML</a>

(3.36)

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M283">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M284">View MathML</a>, it follows from (3.28), (3.31), (3.32), (3.33), and (3.36) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M285">View MathML</a>

(3.37)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M286">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M288">View MathML</a>

(3.38)

Therefore, from (3.34), (3.37), and (3.38), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M289">View MathML</a>

(3.39)

By (3.30) and (3.31), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M290">View MathML</a>

(3.40)

By (1.6), (3.16), (3.35), (3.39), (3.40), and Lemma 2.5, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M292">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M293">View MathML</a>

(3.41)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M294">View MathML</a>, we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M296">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M297">View MathML</a>

which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M298">View MathML</a> of critical values with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M299">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M300">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M301">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M302">View MathML</a> . If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M303">View MathML</a> is bounded, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M304">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M305">View MathML</a>

(3.42)

In a similar fashion to the proof of (3.4) and (3.5), for the given δ in (3.16), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M306">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M307">View MathML</a>

(3.43)

Hence, by (1.6), (2.1), (3.16), (3.42), and (3.43), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M308">View MathML</a>

which, together with (3.42), implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M309">View MathML</a>

This contradicts the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M298">View MathML</a> is unbounded, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M303">View MathML</a> is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

4 Examples

Example 4.1 Consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M312">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M317">View MathML</a>, and a satisfies (A). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M318">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M322">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M323">View MathML</a>

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M325">View MathML</a>. Hence, problem (4.1) has an unbounded sequence of fast homoclinic solutions.

Example 4.2 Consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M326">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M317">View MathML</a>, and a satisfies (A). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M332">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M334">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M335">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M336">View MathML</a>

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/89/mathml/M325">View MathML</a>. Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of fast homoclinic solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Qiongfen Zhang was supported by the NNSF of China (No. 11301108), Guangxi Natural Science Foundation (No. 2013GXNSFBA019004) and the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093). Qi-Ming Zhang was supported by the NNSF of China (No. 11201138). Xianhua Tang was supported by the NNSF of China (No. 11171351).

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