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Solvability of a second-order Hamiltonian system with impulsive effects

Binxiang Dai* and Jia Guo

Author Affiliations

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, China

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Boundary Value Problems 2013, 2013:151  doi:10.1186/1687-2770-2013-151

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/151


Received:24 April 2013
Accepted:30 May 2013
Published:25 June 2013

© 2013 Dai and Guo; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, a class of second-order Hamiltonian systems with impulsive effects are considered. By using critical point theory, we obtain some existence theorems of solutions for the nonlinear impulsive problem. We extend and improve some recent results.

MSC: 334B18, 34B37, 58E05.

Keywords:
Hamiltonian system; impulsive; critical point theory

1 Introduction and main results

Consider the second-order Hamiltonian systems with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>) are continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M7">View MathML</a> satisfies the following assumption:

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8">View MathML</a> is measurable in t for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M9">View MathML</a> and continuously differentiable in x for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M13">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M15">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>), (1.1) is the Hamiltonian system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M19">View MathML</a>

(1.2)

In the past years, the existence of solutions for the second-order Hamiltonian systems (1.2) has been studied extensively via modern variational methods by many authors (see [1-13]).

When the gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M20">View MathML</a> is bounded, that is, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M21">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M22">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>, Mawhin-Willem in [1] proved the existence of solutions for problem (1.2) under the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M25">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M26">View MathML</a>

When the gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M20">View MathML</a> is bounded sublinearly, that is, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M30">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>, Tang [2] proved the existence of solutions for problem (1.2) under the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M33">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M34">View MathML</a>

which generalizes Mawhin-Willem’s results.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M35">View MathML</a>, problem (1.1) gives less results (see [14-16]). In [14], Zhou and Li extended the results of [2] to impulsive problem (1.1); they proved the following theorems.

Theorem A[14]

Assume that (A) and the following conditions are satisfied:

(h1) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M37">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M38">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M15">View MathML</a>.

(h2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M41">View MathML</a>.

(h3) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M44">View MathML</a>

Then problem (1.1) has at least one weak solution.

Theorem B[14]

Suppose that (A) and the condition (h1) of Theorem A hold. Assume that:

(h4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M45">View MathML</a>.

(h5) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M48">View MathML</a>

(h6) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M49">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M50">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M51">View MathML</a>

Then problem (1.1) has at least one weak solution.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M52">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M53">View MathML</a> is convex in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54">View MathML</a> (e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M55">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M57">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M58">View MathML</a> (e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M59">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M62">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8">View MathML</a> satisfies the condition (h2) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem A.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M64">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M53">View MathML</a> satisfies that the gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M66">View MathML</a> is Lipschitz continuous and monotone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54">View MathML</a> (e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M68">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M57">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M58">View MathML</a> (e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M72">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M75">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8">View MathML</a> satisfies the condition (h4) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem B.

In this paper, we further study the existence of solutions for impulsive problem (1.1). Our main results are the following theorems.

Theorem 1.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77">View MathML</a>satisfies assumption (A) and the following conditions hold:

(H1) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M37">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M80">View MathML</a>

(1.5)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23">View MathML</a>and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>.

(H2) There exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M56">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M84">View MathML</a>

(1.6)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M85">View MathML</a>.

(H3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M86">View MathML</a>

(1.7)

(H4) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M89">View MathML</a>

(1.8)

Then impulsive problem (1.1) has at least one weak solution.

Remark 1.1 Theorem 1.1 generalizes Theorem A, which is a special case of our Theorem 1.1 corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M90">View MathML</a>.

Example 1.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92">View MathML</a>. Consider the following impulsive problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M93">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M94">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M95">View MathML</a>

which is bounded and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M96">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M99">View MathML</a>. Then all the conditions of Theorem 1.1 are satisfied. According to Theorem 1.1, the above problem has at least one weak solution. However, F does not satisfy the condition (h1) in Theorem A. Therefore, our result improves and generalizes the Theorem A.

Theorem 1.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77">View MathML</a>satisfies assumption (A) and the condition (H1) of Theorem 1.1. Furthermore, assume that

(H5) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M102">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M103">View MathML</a>

(1.9)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M14">View MathML</a>.

(H6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M105">View MathML</a>

(1.10)

(H7) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M108">View MathML</a>

(1.11)

(H8) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M49">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M50">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M111">View MathML</a>

(1.12)

for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M5">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M6">View MathML</a>.

Then impulsive problem (1.1) has at least one weak solution.

Remark 1.2 Theorem 1.2 generalizes Theorem B, which is a special case of our Theorem 1.2 corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M90">View MathML</a>.

Example 1.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92">View MathML</a>. Consider the following impulsive problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M118">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M119">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M120">View MathML</a>

which is bounded from above, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M121">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M126">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M127">View MathML</a>). Then all the conditions of Theorem 1.2 are satisfied. According to Theorem 1.2, the above problem has at least one weak solution. However, F does not satisfy the condition (h4) in Theorem B. Therefore, our result improves and generalizes Theorem B.

Theorem 1.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M129">View MathML</a>satisfy the assumptions (A), (H1), (H2), (H7) and (H8). Furthermore, assume that

(H9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M130">View MathML</a>

(1.13)

uniformly for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>.

Then impulsive problem (1.1) has at least one weak solution.

Example 1.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M92">View MathML</a>. Consider the following impulsive problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M134">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M135">View MathML</a>

Take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M136">View MathML</a>

Then all the conditions of Theorem 1.3 are satisfied. According to Theorem 1.3, the above problem has at least one weak solution. However, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M8">View MathML</a> is neither superquadratic in X nor subquadratic in X.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M138">View MathML</a> with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M139">View MathML</a>

inducing the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M140">View MathML</a>

The corresponding functional ϕ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M142">View MathML</a>

(2.1)

is continuously differentiable and weakly lower semi-continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141">View MathML</a>. For the sake of convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M144">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M146">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M147">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M148">View MathML</a>

(2.2)

Definition 2.1 We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M149">View MathML</a> is a weak solution of (1.1) if the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M150">View MathML</a>

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M151">View MathML</a>.

It is well known that the solutions of impulsive problem (1.1) correspond to the critical point of ϕ.

Definition 2.2[1]

Let X be a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M153">View MathML</a>.

(1) ϕ is said to satisfy the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154">View MathML</a>-condition on X if the existence of a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M155">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M157">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158">View MathML</a> implies that c is a critical value of ϕ.

(2) ϕ is said to satisfy the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> condition on X if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M155">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M161">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M157">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158">View MathML</a> possesses a convergent subsequence in X.

Remark 2.1 It is clear that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> condition implies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154">View MathML</a>-condition for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M153">View MathML</a>.

Lemma 2.1[1]

Ifϕis weakly lower semi-continuous on a reflexive Banach spaceX (i.e., if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M167">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M168">View MathML</a>) and has a bounded minimizing sequence, thenϕhas a minimum onX.

Remark 2.2 The existence of a bounded minimizing sequence will be in particular ensured when ϕ is coercive, i.e., such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M169">View MathML</a>

Lemma 2.2[1]

LetXbe a Banach space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M170">View MathML</a>. Assume thatXsplits into a direct sum of closed subspaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M171">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M172">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M173">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M174">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M175">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M176">View MathML</a>

Then ifϕsatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M154">View MathML</a>-condition, cis a critical value ofϕ.

Lemma 2.3[1]

If the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M178">View MathML</a>converges weakly touin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M178">View MathML</a>converges uniformly touon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M181">View MathML</a>.

Lemma 2.4[1]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M183">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M184">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M185">View MathML</a>

Lemma 2.5There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M186">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M188">View MathML</a>

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M183">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M190">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M192">View MathML</a>.

3 Proof of main results

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M195">View MathML</a>.

Proof of Theorem 1.1 It follows from (H1) and Sobolev’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M196">View MathML</a>

(3.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a> and some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M200">View MathML</a>. By (H2) and Wirtinger’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M201">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a>.

From (H4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M203">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M182">View MathML</a>. Therefore we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M205">View MathML</a>

(3.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M206">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M207">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M208">View MathML</a>, (3.3) and (H3) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M209">View MathML</a>

By Lemma 2.1 and Remark 2.1, ϕ has a minimum point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M141">View MathML</a>, which is a critical point of ϕ. Therefore, we complete the proof of Theorem 1.1. □

Lemma 3.1Assume that the conditions of Theorem 1.2 hold. Thenϕsatisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a>condition.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M212">View MathML</a> be bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M213">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158">View MathML</a>. From (H1) and Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M215">View MathML</a>

(3.4)

for all large n and some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M218">View MathML</a>. It follows from (H5) and Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M219">View MathML</a>

(3.5)

By (3.4), (3.5), (H8) and Young’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M220">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M222">View MathML</a>. From Wirtinger’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M223">View MathML</a>

(3.7)

The inequalities (3.6) and (3.7) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M224">View MathML</a>

(3.8)

for all large n and some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M226">View MathML</a>. It follows from (H5), Cauchy-Schwarz’s inequality and Wirtinger’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M227">View MathML</a>

(3.9)

Like in the proof of Theorem 1.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M228">View MathML</a>

(3.10)

From (H7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M229">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M230">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M212">View MathML</a> is bounded, from (3.9) and (3.10), there exists a constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M232">View MathML</a>

(3.11)

for all large n and some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M233">View MathML</a>. By the above inequality and (H6), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M234">View MathML</a> is bounded. In fact, if not, without loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M235">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158">View MathML</a>. Then, from (3.8) and the above inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M237">View MathML</a>

which contradicts (H6). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M234">View MathML</a> is bounded. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M239">View MathML</a> is bounded from (3.7) and (3.8). Hence, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M240">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M241">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M242">View MathML</a>

By Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M243">View MathML</a>

On the other hand, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M244">View MathML</a>

(3.12)

It follows from the above equality, (A) and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M245">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M246">View MathML</a>

Thus, we conclude that ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> condition. □

Now, we give the proof of our Theorem 1.2.

Proof of Theorem 1.2 Let W be the subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M248">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M249">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M250">View MathML</a>. Firstly, we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M251">View MathML</a>

(3.13)

In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M252">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M253">View MathML</a>, by the proof of Theorem 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M254">View MathML</a>

(3.14)

Like in the proof of Lemma 3.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M255">View MathML</a>

(3.15)

By (H8) and Lemma 2.5, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M256">View MathML</a>

(3.16)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M257">View MathML</a>. By (3.14), (3.15) and (3.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M258">View MathML</a>

(3.17)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M257">View MathML</a>.

By Lemma 2.4, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M260">View MathML</a> on W. Hence (3.13) follows from (3.17).

On the other hand, by (H7), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M261">View MathML</a>

(3.18)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M262">View MathML</a>. Therefore, from (3.18) and (H6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M263">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M264">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M54">View MathML</a>. It follows from Lemma 2.2 and Lemma 3.1 that problem (1.1) has at least one weak solution. □

Proof of Theorem 1.3 First we prove that ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> condition. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M267">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> sequence for ϕ, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M213">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M158">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M271">View MathML</a> is bounded. In a way similar to the proof of Theorem 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M272">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M273">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M262">View MathML</a>. Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M275">View MathML</a>

From (3.7) and the above inequalities, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M276">View MathML</a>

(3.19)

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M278">View MathML</a>.

By (H9) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M279">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M280">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M283">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>. It follows from assumption (A) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M286">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M287">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M289">View MathML</a>.

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M290">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M291">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M10">View MathML</a>.

By the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M271">View MathML</a>, (H7) and (3.19), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M296">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M297">View MathML</a> is bounded. Like in the proof of Lemma 3.1, we know that ϕ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/151/mathml/M159">View MathML</a> condition.

Furthermore, we can prove Theorem 1.3 using the same way as in the proof of Theorem 1.2. Here, we omit it. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

We express our gratitude to the referees for their valuable criticism of the manuscript and for helpful suggestions. Supported by the National Natural Science Foundation of China (11271371, 10971229).

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