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Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

Jianwen Zhou1, Yanning Wang2 and Yongkun Li1*

Author affiliations

1 Department of Mathematics, Yunnan University, Kunming, Yunnan, 650091, People’s Republic of China

2 Oxbridge College, Kunming University of Science and Technology, Kunming, Yunnan, 650106, People’s Republic of China

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Citation and License

Boundary Value Problems 2012, 2012:148  doi:10.1186/1687-2770-2012-148

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/148


Received:17 September 2012
Accepted:6 December 2012
Published:21 December 2012

© 2012 Zhou et al.; licensee Springer

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

Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M4">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M6">View MathML</a> is a symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7">View MathML</a> matrix-valued function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M9">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M13">View MathML</a>) are continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M14">View MathML</a>. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.

MSC: 34B37, 34N05.

Keywords:
nonautonomous second-order systems; time scales; impulse; variational approach

1 Introduction

Consider the nonautonomous second-order system on time scales with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M15">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M4">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M6">View MathML</a> is a symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7">View MathML</a> matrix-valued function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M9">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M27">View MathML</a>) are continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M14">View MathML</a> satisfies the following assumption:

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29">View MathML</a> is Δ-measurable in t for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M30">View MathML</a> and continuously differentiable in x for Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M34">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M36">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M37">View MathML</a> denotes the gradient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29">View MathML</a> in x.

For the sake of convenience, in the sequel, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M40">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44">View MathML</a> is a zero matrix, (1.1) is the Hamiltonian system on time scales

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M45">View MathML</a>

(1.2)

In [1], the authors study the Sobolev’s spaces on time scales and their properties. As applications, they present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for (1.2).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44">View MathML</a> is not a zero matrix, until now the variational structure of (1.1) has not been studied.

Problem (1.1) covers the second-order Hamiltonian system with impulsive effects (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M50">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M51">View MathML</a>

(1.3)

as well as the second-order discrete Hamiltonian system (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M54">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M55">View MathML</a>

In [2], the authors establish some sufficient conditions on the existence of solutions of (1.3) by means of some critical point theorems when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M56">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M57">View MathML</a>, until now, it is unknown whether problem (1.1) has a variational structure or not.

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [3-5]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory and so on (see [6-9]).

For a second-order differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M58">View MathML</a>, one usually considers impulses in the position u and the velocity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M59">View MathML</a>. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position (see [10]). The impulses only on the velocity occur also in impulsive mechanics (see [11]). An impulsive problem with impulses in the derivative only is considered in [12].

The study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are Refs. [13-17]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [18,19]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks and social sciences (see [16]). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.

There have been many approaches to study solutions of differential equations on time scales, such as the method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on (see [1,20-29]). In [24], authors used the fixed point theorem of strict-set-contraction to study the existence of positive periodic solutions for functional differential equations with impulse effects on time scales. However, the study of the existence and multiplicity of solutions for differential equations on time scales using the variational method has received considerably less attention (see, for example, [1,29]). The variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems, with some type of discontinuities such as impulses.

Motivated by the above, we research the existence of variational construction for problem (1.1) in an appropriate space of functions and study the existence of solutions for (1.1) by some critical point theorems in this paper. All these results are new.

2 Preliminaries and statements

In this section, we present some fundamental definitions and results from the calculus on time scales and Sobolev’s spaces on time scales that will be required below. These are a generalization to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M60">View MathML</a> of definitions and results found in [17].

Definition 2.1 ([[17], Definition 1.1])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61">View MathML</a> be a time scale. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M62">View MathML</a>, the forward jump operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M63">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M64">View MathML</a>

while the backward jump operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M65">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M66">View MathML</a>

(supplemented by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M68">View MathML</a>, where ∅ denotes the empty set). A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M62">View MathML</a> is called right-scattered, left-scattered, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M71">View MathML</a> hold, respectively. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M73">View MathML</a>, then t is called right-dense, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M75">View MathML</a>, then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76">View MathML</a> which is derived from the time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61">View MathML</a> as follows. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61">View MathML</a> has a left-scattered maximum m, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M79">View MathML</a>; otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M80">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M82">View MathML</a>, we denote the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M85">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M87">View MathML</a>

respectively. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M88">View MathML</a> if b is left-dense and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M89">View MathML</a> if b is left-scattered. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M90">View MathML</a>, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M91">View MathML</a> if b is left-dense and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M92">View MathML</a> if b is left-scattered.

Definition 2.2 ([[17], Definition 1.10])

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M93">View MathML</a> is a function and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94">View MathML</a>. Then we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95">View MathML</a> to be the number (provided it exists) with the property that given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M96">View MathML</a>, there is a neighborhood U of t (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M97">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M99">View MathML</a>

We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95">View MathML</a> the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95">View MathML</a> exists for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M104">View MathML</a> is then called the delta derivative of f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76">View MathML</a>.

Definition 2.3 ([[1], Definition 2.3])

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106">View MathML</a> is a function,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M107">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M94">View MathML</a>. Then we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M109">View MathML</a> (provided it exists). We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95">View MathML</a> the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M95">View MathML</a> exists for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M112">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M113">View MathML</a> is then called the delta derivative of f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M76">View MathML</a>.

Definition 2.4 ([[17], Definition 2.7])

For a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M93">View MathML</a>, we will talk about the second derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M116">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M117">View MathML</a> is differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M118">View MathML</a> with derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M119">View MathML</a>.

Definition 2.5 ([[1], Definition 2.5])

For a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106">View MathML</a>, we will talk about the second derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M116">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M117">View MathML</a> is differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M118">View MathML</a> with derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M124">View MathML</a>.

The Δ-measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M125">View MathML</a> and Δ-integration are defined as those in [26].

Definition 2.6 ([[1], Definition 2.7])

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M106">View MathML</a> is a function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M127">View MathML</a> and let A be a Δ-measurable subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M61">View MathML</a>. f is integrable on A if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M129">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M12">View MathML</a>) are integrable on A, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M131">View MathML</a>.

Definition 2.7 ([[17], Definition 2.3])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M132">View MathML</a>. B is called a Δ-null set if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M133">View MathML</a>. Say that a property P holds Δ-almost everywhere (Δ-a.e.) on B, or for Δ-almost all (Δ-a.a.) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M134">View MathML</a> if there is a Δ-null set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M135">View MathML</a> such that P holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M136">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M138">View MathML</a>, we set the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M139">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M140">View MathML</a>

We have the following theorem.

Theorem 2.1 ([[1], Theorem 2.1])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M138">View MathML</a>. Then the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M143">View MathML</a>is a Banach space together with the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M144">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M145">View MathML</a>is a Hilbert space together with the inner product given for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M146">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M147">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M148">View MathML</a>denotes the inner product in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M149">View MathML</a>.

Definition 2.8 ([[1], Definition 2.11])

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M150">View MathML</a>. We say that f is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M152">View MathML</a>) if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M96">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M155">View MathML</a> is a finite pairwise disjoint family of subintervals of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M83">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M157">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M158">View MathML</a>.

Now, we recall the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M8">View MathML</a> defined in [1]. For the sake of convenience, in the sequel we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M161">View MathML</a>.

Definition 2.9 ([[1], Definition 2.12])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M164">View MathML</a>. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M166">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M167">View MathML</a> such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M168">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M169">View MathML</a>

(2.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M172">View MathML</a>

It follows from Remark 2.2 in [1] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M173">View MathML</a>

is true for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163">View MathML</a>. These two sets are, as a class of functions, equivalent. It is the characterization of functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a> in terms of functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M177">View MathML</a> too. That is the following theorem.

Theorem 2.2 ([[1], Theorem 2.5])

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163">View MathML</a>, and that (2.1) holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M168">View MathML</a>. Then there exists a unique function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M182">View MathML</a>such that the equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M183">View MathML</a>

(2.2)

are satisfied and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M184">View MathML</a>

(2.3)

By identifying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M165">View MathML</a> with its absolutely continuous representative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M186">View MathML</a> for which (2.2) holds, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a> can be endowed with the structure of a Banach space. That is the following theorem.

Theorem 2.3 ([[25], Theorem 2.21])

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M137">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M163">View MathML</a>. The set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a>is a Banach space together with the norm defined as

(2.4)

Moreover, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M192">View MathML</a>is a Hilbert space together with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M193">View MathML</a>

The Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a> has some important properties.

Theorem 2.4 ([[25], Theorem 2.23])

There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M195">View MathML</a>such that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M196">View MathML</a>

(2.5)

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M198">View MathML</a>.

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M199">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M200">View MathML</a>

Theorem 2.5 ([[25], Theorem 2.25])

If the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M201">View MathML</a>converges weakly touin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M202">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M203">View MathML</a>converges strongly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M204">View MathML</a>tou.

Theorem 2.6 ([[25], Theorem 2.27])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M205">View MathML</a>be Δ-measurable intfor each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M206">View MathML</a>and continuously differentiable in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M207">View MathML</a>for Δ-almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M210">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M211">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M212">View MathML</a>) such that for Δ-almost<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>and every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M206">View MathML</a>, one has

(2.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M216">View MathML</a>, then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M217">View MathML</a>defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M218">View MathML</a>

is continuously differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M159">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M220">View MathML</a>

(2.7)

3 Variational setting

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1.1) to the one of seeking the critical points of a corresponding functional.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>, by identifying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a> with its absolutely continuous representative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M223">View MathML</a> for which (2.2) holds, then u is absolutely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M224">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M225">View MathML</a> may not hold for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M226">View MathML</a>. This leads to impulsive effects.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M227">View MathML</a> and multiply the two sides of the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M228">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M229">View MathML</a> and integrate on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M230">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M231">View MathML</a>

(3.1)

Moreover, combining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M232">View MathML</a>, one has

Combining (3.1), we have

Considering the above, we introduce the following concept solution for problem (1.1).

Definition 3.1 We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a> is a weak solution of problem (1.1) if the identity

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M227">View MathML</a>.

Consider the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M238">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M239">View MathML</a>

(3.2)

where

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M241">View MathML</a>

Lemma 3.1The functionalφis continuously differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M243">View MathML</a>

(3.3)

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M244">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M247">View MathML</a> satisfies all assumptions of Theorem 2.6. Hence, by Theorem 2.6, we know that the functional ψ is continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M249">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M250">View MathML</a>.

On the other hand, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M253">View MathML</a>, one has that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M254">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M255">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M250">View MathML</a>. Thus, φ is continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> and (3.3) holds. □

By Definition 3.1 and Lemma 3.1, the weak solutions of problem (1.1) correspond to the critical points of φ.

Moreover, we need more preliminaries. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M259">View MathML</a>

We see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M260">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M261">View MathML</a> is the bounded self-adjoint linear operator defined, using the Riesz representation theorem, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M262">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M263">View MathML</a> and I denote an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M7">View MathML</a> identity matrix and an identity operator, respectively. By (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M265">View MathML</a> can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M266">View MathML</a>

(3.4)

The compact imbedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M268">View MathML</a> implies that K is compact. By classical spectral theory, we can decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> into the orthogonal sum of invariant subspaces for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M271">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M272">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M274">View MathML</a> are such that, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M98">View MathML</a>,

(3.5)

(3.6)

Remark 3.1K has only finitely many eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M278">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M279">View MathML</a> since K is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M280">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M273">View MathML</a> is finite dimensional. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270">View MathML</a> is a compact perturbation of the self-adjoint operator I. By a well-known theorem, we know that 0 is not in the essential spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M270">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M284">View MathML</a> is a finite dimensional space too.

To prove our main results, we need the following definitions and theorems.

Definition 3.2 ([[30], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M285">View MathML</a>])

Let X be a real Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286">View MathML</a>. I is said to be satisfying (PS) condition on X if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M287">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M288">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M289">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M290">View MathML</a>, possesses a convergent subsequence in X.

Firstly, we state the local linking theorem.

Let X be a real Banach space with a direct decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M291">View MathML</a>. Consider two sequences of a subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M292">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M293">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M294">View MathML</a>

For every multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M295">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M296">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M297">View MathML</a>. We say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M298">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M299">View MathML</a>. A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M300">View MathML</a> is admissible if, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M301">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M302">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M303">View MathML</a>.

Definition 3.3 ([[31], Definition 2.2])

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305">View MathML</a> condition if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M306">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M307">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M308">View MathML</a>

contains a subsequence which converges to a critical point of I.

Theorem 3.1 [[31], Theorem 2.2]

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286">View MathML</a>satisfies the following assumptions:

(I1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M310">View MathML</a>andIhas a local linking at 0 with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M311">View MathML</a>; that is, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M312">View MathML</a>,

(I2) Isatisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305">View MathML</a>condition.

(I3) Imaps bounded sets into bounded sets.

(I4) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M316">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M318">View MathML</a>.

ThenIhas at least two critical points.

Remark 3.2 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M286">View MathML</a>, by the condition (I1) of Theorem 3.1, 0 is the critical point of I. Thus, under the conditions of Theorem 3.1, I has at least one nontrivial critical point.

Secondly, we state another three critical point theorems.

Theorem 3.2 ([[32], Theorem 5.29])

LetEbe a Hilbert space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M320">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M321">View MathML</a>. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M322">View MathML</a>, satisfies (PS) condition, and

(I5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M323">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M324">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M325">View MathML</a>is bounded and self-adjoint, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M326">View MathML</a>,

(I6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M327">View MathML</a>is compact, and

(I7) there exist a subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328">View MathML</a>and sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M329">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M330">View MathML</a>and constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M331">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M332">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M333">View MathML</a>,

(ii) Qis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M334">View MathML</a>,

(iii) Sand∂Qlink.

ThenIpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M335">View MathML</a>.

Theorem 3.3 ([[32], Theorem 9.12])

LetEbe a Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M336">View MathML</a>be an even functional which satisfies the (PS) condition and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M337">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M338">View MathML</a>, whereVis finite dimensional, andIsatisfies

(I8) there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M339">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M340">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M341">View MathML</a>,

(I9) for each finite dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M342">View MathML</a>, there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M343">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M344">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M345">View MathML</a>,

thenIpossesses an unbounded sequence of critical values.

In order to state another critical point theorem, we need the following notions. Let X and Y be Banach spaces with X being separable and reflexive, and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M347">View MathML</a> be a dense subset. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M348">View MathML</a>, there is a semi-norm on E defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M349">View MathML</a>

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350">View MathML</a> the topology on E induced by a semi-norm family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M351">View MathML</a>, and let w and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M352">View MathML</a> denote the weak-topology and weak*-topology, respectively.

For a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M353">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M354">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M355">View MathML</a> is said to be weak sequentially continuous if, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M356">View MathML</a> in E, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M357">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M358">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M359">View MathML</a> is sequentially continuous. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M360">View MathML</a>, we say that Φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M361">View MathML</a> condition if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M362">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M363">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M364">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M365">View MathML</a> contains a convergent subsequence.

Suppose that

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366">View MathML</a>) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M360">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M368">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350">View MathML</a>-closed, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M370">View MathML</a> is continuous;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M371">View MathML</a>) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M372">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M373">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M374">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375">View MathML</a>) there exist a finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M376">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M377">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M378">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M379">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M380">View MathML</a>

Theorem 3.4 ([33])

Assume that Φ is even and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375">View MathML</a>) are satisfied. Then Φ has at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M383">View MathML</a>pairs of critical points with critical values less than or equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M384">View MathML</a>provided Φ satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M361">View MathML</a>condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M386">View MathML</a>.

Remark 3.3 In our applications, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M387">View MathML</a>=<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M388">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M350">View MathML</a> is the product topology on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346">View MathML</a> given by the weak topology on X and the strong topology on Y.

4 Main results

Lemma 4.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M391">View MathML</a>is compact on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M393">View MathML</a> be any bounded sequence. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> is a Hilbert space, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M356">View MathML</a>. Theorem 2.5 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M396">View MathML</a>. By (2.5), we have

The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251">View MathML</a> and this imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M399">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. The proof is complete. □

First of all, we give two existence results.

Theorem 4.1Suppose that (A) and the following conditions are satisfied.

(F1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M401">View MathML</a>uniformly for Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>,

(F2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M403">View MathML</a>uniformly for Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>,

(F3) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M405">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M406">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M407">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M408">View MathML</a>

(F4) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M409">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M410">View MathML</a>

(F5) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M411">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M412">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M413">View MathML</a>

(F6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M414">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M415">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M417">View MathML</a>,

(F7) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M418">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M419">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M420">View MathML</a>

Then problem (1.1) has at least two weak solutions. The one is a nontrivial weak solution, the other is a trivial weak solution.

Proof By Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M421">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M422">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M423">View MathML</a> being its Hilbertian basis, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M424">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M425">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M426">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M427">View MathML</a>

We divide our proof into four parts in order to show Theorem 4.1.

Firstly, we show that φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305">View MathML</a> condition.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M307">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M432">View MathML</a>

then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M433">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M434">View MathML</a>

(4.1)

for all large n. On the other hand, by (F3), there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M435">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M436">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M437">View MathML</a>

(4.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M438">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. By (A) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M440">View MathML</a>

(4.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M441">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. It follows from (4.2) and (4.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M443">View MathML</a>

(4.4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M446">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M10">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M448">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M449">View MathML</a>

(4.5)

From (F5) and (2.5), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M450">View MathML</a>

(4.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M452">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M453">View MathML</a>. Combining (4.4), (4.5), (4.6) and Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M454">View MathML</a>

(4.7)

for all large n, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M455">View MathML</a>. On the other hand, by (F3), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M457">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M458">View MathML</a>

(4.8)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M459">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. By (A),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M461">View MathML</a>

(4.9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M462">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M464">View MathML</a>. Combining (4.8) and (4.9), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M465">View MathML</a>

(4.10)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. According to (F7), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M468">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M469">View MathML</a>

(4.11)

Thus, by (4.1), (4.10) and (4.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M470">View MathML</a>

(4.12)

for all large n. From (4.12), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M471">View MathML</a> is bounded. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M472">View MathML</a>, by Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M473">View MathML</a>

(4.13)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M474">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M253">View MathML</a>, by (4.7) and (4.13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M479">View MathML</a>, by (2.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M480">View MathML</a>

(4.14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M412">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M482">View MathML</a>, by (4.7) and (4.14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429">View MathML</a> is also bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M429">View MathML</a> is also bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. Going if necessary to a subsequence, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M487">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. From Theorem 2.5, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M489">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M490">View MathML</a>. Since

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M492">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M493">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M494">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. Hence φ satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M305">View MathML</a> condition.

Secondly, we show that φ maps bounded sets into bounded sets.

It follows from (3.2), (4.4), (4.5) and (4.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M497">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>. Thus, φ maps bounded sets into bounded sets.

Thirdly, we claim that φ has a local linking at 0 with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M311">View MathML</a>.

Applying (F2), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M500">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M501">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M502">View MathML</a>

(4.15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M503">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. By (F7), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M505">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M506">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M507">View MathML</a>

(4.16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M508">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M509">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M510">View MathML</a>, by (2.5), (3.2), (3.6), (4.15) and (4.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M511">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M512">View MathML</a>

On the other hand, it follows from (F6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M513">View MathML</a>

(4.17)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M197">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M515">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M516">View MathML</a>. Using (F4), (2.5), (3.2), (3.5) and (4.17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M517">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M518">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M519">View MathML</a>. Then φ satisfies the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M520">View MathML</a> of Theorem 3.1.

Finally, we claim that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M522">View MathML</a>

For given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M524">View MathML</a> is a finite dimensional space, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M525">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M526">View MathML</a>

(4.18)

By (F1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M527">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M528">View MathML</a>

(4.19)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M529">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. From (A), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M531">View MathML</a>

(4.20)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M532">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>. Equations (4.19) and (4.20) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M534">View MathML</a>

(4.21)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35">View MathML</a> and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M208">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M537">View MathML</a>. Using (3.2), (3.6), (4.5), (4.17), (4.18) and (4.21), we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M538">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M539">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M540">View MathML</a>. Hence, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M542">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M317">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M524">View MathML</a>.

Thus, by Theorem 3.1, problem (1.1) has at least one nontrivial weak solution. The proof is complete. □

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M546">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M547">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M548">View MathML</a>. Consider the second-order Hamiltonian system with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M549">View MathML</a>

(4.22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M550">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M551">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M552">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M553">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M554">View MathML</a>

then all conditions of Theorem 4.1 hold. According to Theorem 4.1, problem (4.22) has at least one nontrivial weak solution. In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M555">View MathML</a>

is the solution of problem (4.22).

Theorem 4.2Assume that (A), (F5), (F6), (F7) and the following conditions are satisfied.

(F8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M556">View MathML</a>uniformly for Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M557">View MathML</a>,

(F9) there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M559">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M560">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M562">View MathML</a>,

(F10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M563">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M35">View MathML</a>and Δ-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M31">View MathML</a>.

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

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M566">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M567">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M568">View MathML</a>. Then E is a real Hilbert space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M571">View MathML</a>.

Firstly, we prove that φ satisfies the (PS) condition. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M393">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M573">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M574">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M365">View MathML</a>. As the proof of Theorem 4.1, it suffices to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M576">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>. By (F9) there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M578">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M579">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M580">View MathML</a>

(4.23)

(see [34]). By (F9), (4.11) and (4.23), we have

(4.24)

for large k, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M582">View MathML</a>. Equation (4.24) implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M583">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M584">View MathML</a>

(4.25)

Combining (3.2), (4.6), (4.11) and (4.25), we obtain

(4.26)

for large k. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M474">View MathML</a>, by (4.26), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M576">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M242">View MathML</a>.

For any small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M590">View MathML</a>, by (F8) we know that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M591">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M592">View MathML</a>

(4.27)

By (F7), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M593">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M594">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M595">View MathML</a>

(4.28)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M596">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M597">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M598">View MathML</a>, by (2.5), (3.2), (3.6), (4.27) and (4.28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M599">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M600">View MathML</a>

(4.29)

Moreover, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M601">View MathML</a> is compact (see [[35], p.1437]). It follows from (3.4), (4.29) and Lemma 4.1 that φ satisfies the conditions (I5), (I6) and (I7)(i) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M602">View MathML</a> of Theorem 3.2.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M603">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M604">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M605">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M606">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M607">View MathML</a>. Then S and ∂Q link, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M608">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M609">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M610">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M611">View MathML</a>.

By (F10), (3.4), (3.5) and (4.17), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M612">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M613">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M614">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M615">View MathML</a>. By the equivalence of a finite dimensional space and (4.23), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M616">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M617">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M618">View MathML</a>

for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M604">View MathML</a> due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M558">View MathML</a>.

Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M621">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M623">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M624">View MathML</a>. By the equivalence of a finite dimensional space and (4.23), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M625">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M626">View MathML</a>

for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M627">View MathML</a>.

Summing up the above, φ satisfies all conditions of Theorem 3.2. Hence, φ possesses a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M628">View MathML</a>, and hence problem (1.1) has at least one nontrivial weak solution. The proof is complete. □

Remark 4.1 There are a number of functions satisfying (A), (F8), (F9) and (F10), for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M629">View MathML</a>.

Next, we given two multiplicity results.

Theorem 4.3Assume that (A), (F5), (F7), (F8), (F9) and the following conditions are satisfied.

(F11) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M251">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M43">View MathML</a>) are odd.

(F12) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29">View MathML</a>is even inxand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M634">View MathML</a>.

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

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M635">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M636">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M637">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M639">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M640">View MathML</a>. From the proof of Theorem 4.2, we know that φ satisfies the (PS) condition, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M641">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M642">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M643">View MathML</a>

For each finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328">View MathML</a>, combining (3.2), (4.5), (4.6), (4.23) and the equivalence of a finite dimensional space, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M645">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M646">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M647">View MathML</a>

(4.30)

This implies that there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M648">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M649">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M650">View MathML</a>.

Moreover, by (F10) and (F12), we know that φ is even and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M651">View MathML</a>. In view of Theorem 3.3, φ has a sequence of critical points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M652">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M653">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M654">View MathML</a> is bounded in E, then by the definition of φ, one knows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M655">View MathML</a> is also bounded, a contradiction. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M654">View MathML</a> is unbounded in E. The proof is completed. □

Example 4.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M657">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M658">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M659">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M660">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M661">View MathML</a>. Consider the second-order Hamiltonian system with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M662">View MathML</a>

(4.31)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M44">View MathML</a> is the unit matrix and

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M665">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M666">View MathML</a>. All conditions of Theorem 4.3 hold. According to Theorem 4.3, problem (4.31) has an unbounded sequence of weak solutions.

Remark 4.2 In Theorem 4.3, if we delete the condition ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M667">View MathML</a>’, we have the following theorem.

Theorem 4.4Assume that (A), (F5), (F7), (F8), (F9), (F11) and the following condition are satisfied.

(F13) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M29">View MathML</a>is even inx.

Then problem (1.1) has an infinite sequence of distinct weak solutions.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M669">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M670">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M637">View MathML</a> in Theorem 3.4. Then, from the proof of Theorem 4.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M346">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M673">View MathML</a>, φ is even, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M640">View MathML</a> satisfies the (PS) condition, and there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M675">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M676">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M677">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M678">View MathML</a>.

For each finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M328">View MathML</a>, by (4.30), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M680">View MathML</a>

Consequently, for each finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M376">View MathML</a>, the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M375">View MathML</a>) holds. Moreover, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M673">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M684">View MathML</a>, we know that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/148/mathml/M366">View MathML</a>) holds too. Therefore, the conclusion follows from Theorem 2.6. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183, the Natural Sciences Foundation of Yunnan Province (2011Y116, 2012FB111, IRTSTYN) and the third batch young skeleton teachers training plan of Yunnan University (XT412003).

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