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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions

Peng Zhang and Yanping Gong*

Author Affiliations

Business School, Central South University, Changsha, Hunan, 410083, P.R. China

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Boundary Value Problems 2012, 2012:82  doi:10.1186/1687-2770-2012-82

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/82


Received:7 May 2012
Accepted:18 July 2012
Published:31 July 2012

© 2012 Zhang and Gong; 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 study the existence and multiplicity results of solutions for some class of fractional differential inclusions with boundary conditions. Some existence and multiplicity results of solutions are given by using the least action principle and minmax methods in nonsmooth critical point theory. Recent results in the literature are generalized and improved. Some examples are given in the paper to illustrate our main results.

MSC: 26A33, 26A42, 58E05, 70H05.

Keywords:
fractional differential inclusions; nonsmooth critical point theory; boundary value problem; variational methods

1 Introduction

In this paper, we consider the fractional boundary value problem (BVP for short) for the following differential inclusion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M4">View MathML</a> are the left and right Riemann-Liouville fractional integrals of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M5">View MathML</a> respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M6">View MathML</a> satisfies the following assumptions:

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M7">View MathML</a> is measurable in t for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M8">View MathML</a> and locally Lipschitz in x for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M10">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M13">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M15">View MathML</a>.

Differential equations with fractional order are generalization of ordinary differential equations to non-integer order. Fractional differential equations have received increasing attention during recent years, since the behavior of physical systems can be properly described by using fractional order system theory. So fractional differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs and articles of Kilbas et al.[1], Miller and Ross [2], Podlubny [3], Samko et al.[4], Agarwal [5], Lakshmikantham [6] and Vasundhara Devi [7] and the references therein.

Recently, fractional differential equations have been of great interest, and boundary value problems for fractional differential equations have been considered by the use of techniques of nonlinear analysis (fixed-point theorems [8-10], Leray-Schauder theory [11,12], lower and upper solution method, monotone iterative method [13-15]).

Variational methods have turned out to be a very effective analytical tool in the study of nonlinear problems. The classical critical point theory for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M16">View MathML</a> functional was developed in the sixties and seventies (see [16,17]). The celebrated and important result in the last 30 years was the mountain pass theorem due to Ambrosetti and Rabinowitz [18] in 1973. The needs of specific applications (such as nonsmooth mechanics, nonsmooth gradient systems, etc.) and the impressive progress in nonsmooth analysis and multivalued analysis led to extensions of the critical point theory to nondifferentiable functions, locally Lipschitz functions in particular. The nonsmooth critical point theory for locally Lipschitz functions started with the work of Chang (see [19]). The theory of Chang was based on the subdifferential of locally Lipschitz functionals due to Clarke (see [20]). Using this subdifferential, Chang proposed a generalization of the well-known Palais-Smale condition and obtained various minimax principles concerning the existence and characterization of critical points for locally Lipschitz functions. Chang used his theory to study semilinear elliptic boundary value problems with a discontinuous nonlinearity. Later, in 2000, Kourogenis and Papageorgiou (see [21]) extended the theory of Chang and obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. Subsequently, many authors also studied the nonsmooth critical point theory (see [22-26]), then the nonsmooth critical point theory is also widely used to deal with nonlinear boundary value problems (see [27-31]). A good survey for nonsmooth critical point theory and nonlinear boundary value problems is the book of Gasinski and Papageorgiou [32].

There are some papers which are devoted to the boundary value problems for fractional differential inclusions (see [33-35]), and the main tools they use are fixed point theory for multi-valued contractions. However, to the best of the authors’ knowledge, there are few results on the solutions to fractional BVP which were established by the nonsmooth critical point theory, since it is often very difficult to establish a suitable space and variational functional for fractional differential equations with boundary conditions. Recently, Jiao and Zhou [36] introduced some appropriate function spaces as their working space and set up a variational functional for the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M17">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M4">View MathML</a> are the left and right Riemann-Liouville fractional integrals of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M5">View MathML</a> respectively, and F is continuously differentiable.

They give two existence results of solutions for the above system by using the least action principle and mountain pass theorem in critical point theory. It is easy to see that system (1.1) is a generalization to system (1.2), and it is interesting to ask whether the results in [36] hold true when the potential F is just locally Lipschitz. But the main difficulty is the variational structure given in [36] cannot be applied to system (1.1) directly. So we have to find a new approach to solve this problem, and the main idea of the new approach comes from the inspiration of Theorem 2.7.3 and Theorem 2.7.5 in [20].

The structure of the paper is as follows. In the next section, for the convenience of readers, we present the mathematical background needed and the corresponding variational structure for system (1.1). In Section 3, using variational methods, we prove two existence theorems for the solutions of problem (1.1) which generalize the results in [36]. Finally, in Section 4, two examples are presented to illustrate our results.

2 Preliminaries

Definition 2.1 (Left and right Riemann-Liouville fractional integrals)

Let f be a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M22">View MathML</a>. The left and right Riemann-Liouville fractional integrals of order γ for function f, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M24">View MathML</a> respectively, are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M25">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M26">View MathML</a>

provided the right-hand sides are pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27">View MathML</a>, where Γ is the gamma function.

Definition 2.2 (Left and right Riemann-Liouville fractional derivatives) Let f be a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M22">View MathML</a>. The left and right Riemann-Liouville fractional derivatives of order γ for function f, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M30">View MathML</a> respectively, are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M31">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M35">View MathML</a>.

Definition 2.3 (Left and right Caputo fractional derivatives)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M35">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M39">View MathML</a>, then the left and right Caputo fractional derivatives of order γ for function f, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M41">View MathML</a> respectively, exist almost everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M44">View MathML</a> are represented by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M45">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M46">View MathML</a>

respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33">View MathML</a>. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M48">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M49">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M50">View MathML</a>

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M52">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M54">View MathML</a> are represented by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M55">View MathML</a>

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33">View MathML</a>.

Property 2.1 ([36])

The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M58">View MathML</a>

at any point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M33">View MathML</a> for a continuous function f and for almost every point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M27">View MathML</a> if the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M61">View MathML</a>.

Definition 2.4 ([36])

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63">View MathML</a>. The fractional derivative space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a> is defined by the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M65">View MathML</a> with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M66">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M65">View MathML</a> denotes the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M68">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69">View MathML</a>. It is obvious that the fractional derivative space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a> is the space of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M71">View MathML</a> having an α-order Caputo fractional derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69">View MathML</a>.

Proposition 2.1 ([36])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63">View MathML</a>. The fractional derivative space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a>is a reflexive and separable Banach space.

Proposition 2.2 ([36])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M63">View MathML</a>. For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M79">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M80">View MathML</a>

(2.1)

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M81">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M82">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M83">View MathML</a>

(2.2)

According to (2.1), we can consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a> with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M85">View MathML</a>

(2.3)

Proposition 2.3 ([36])

Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M87">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M81">View MathML</a>and the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89">View MathML</a>converges weakly touin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M93">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M94">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>.

In this paper, we treat BVP (1.1) in the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M96">View MathML</a> with the equivalent norm defined in (2.3).

Proposition 2.4 ([36])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M97">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M99">View MathML</a>

In order to establish the variational structure for system (1.1), it is necessary to construct some appropriate function spaces. The Cartesian product space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M100">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M101">View MathML</a>

is also a reflexive and separable Banach space with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M102">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M103">View MathML</a>.

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M104">View MathML</a> is a closed subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M105">View MathML</a> under the norm (2.4) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M64">View MathML</a> is closed by Definition 2.4.

In this paper, we use the norm defined in (2.3), which is an equivalent norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M107">View MathML</a> with norm (2.4).

Definition 2.5 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108">View MathML</a> denote the space of essentially bounded measurable functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110">View MathML</a> under the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M111">View MathML</a>

(2.5)

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108">View MathML</a> is a Banach space under the norm (2.5).

Remark 2.5 We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M114">View MathML</a> to denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M116">View MathML</a> respectively.

Definition 2.6 ([13])

Let f be Lipschitz near a given point x in a Banach space X, and v be any other vector in X. The generalized directional derivative of f at x in the direction v, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M117">View MathML</a>, is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M118">View MathML</a>

where y is also a vector in X and λ is a positive scalar, and we denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M119">View MathML</a>

the generalized gradient of f at x (the Clarke subdifferential).

Lemma 2.1 ([20])

Letxandybe points in a Banach spaceX, and suppose thatfis Lipschitz on an open set containing the line segment<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M120">View MathML</a>. Then there exists a pointuin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M121">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M122">View MathML</a>

Definition 2.7 ([32])

A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M123">View MathML</a> is said to be a critical point of a locally Lipschitz f if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M124">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M125">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M126">View MathML</a>. A real number c is called a critical value of f if there is a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M123">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M128">View MathML</a>.

Definition 2.8 ([32])

If f is a locally Lipschitz function, we say that f satisfies the nonsmooth (P.S.) condition if each sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M129">View MathML</a> in X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M130">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M131">View MathML</a> has a convergent subsequence, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M132">View MathML</a>.

Clarke considered the following abstract framework in [20]:

• let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M133">View MathML</a> be a σ-finite positive measure space, and let Y be a separable Banach space;

• let Z be a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134">View MathML</a> denotes the space of measure essentially bounded functions mapping S to Y, equipped with the usual supremum norm;

• define a functional f on Z via

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M136">View MathML</a>

where Z is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M138">View MathML</a> is a given family of functions;

• suppose that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M139">View MathML</a> is measurable for each v in Y, and that x is a point at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M140">View MathML</a> is defined (finitely);

• suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M141">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M142">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M143">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M144">View MathML</a>

(2.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M145">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M147">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M148">View MathML</a>.

Under the conditions described above, f is Lipschitz in a neighborhood of x and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M149">View MathML</a>

(2.7)

Further, if each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M150">View MathML</a> is regular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M151">View MathML</a> for each t, then f is regular at x and the equality holds.

Remark 2.6 The interpretation of (2.7) is as follows: To every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M152">View MathML</a>, there is a corresponding mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M153">View MathML</a> from S to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M154">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M155">View MathML</a>

and having the property that for every v in Z, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M156">View MathML</a>

Thus, every ζ in the left-hand side of (2.7) is an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M157">View MathML</a> that can be written

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M158">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M159">View MathML</a> is a measurable selection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M160">View MathML</a>.

Lemma 2.2LetFsatisfy the condition (A) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M161">View MathML</a>be given by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M162">View MathML</a>, then define a functionalfon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M164">View MathML</a>

Thenfis Lipschitz on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a>, and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M166">View MathML</a>

(2.8)

Proof Take an arbitrary element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M167">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a>, then it suffices to prove f is Lipschitz on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M167">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M170">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M171">View MathML</a>), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M172">View MathML</a>

(2.9)

by Proposition 2.2, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M173">View MathML</a>. In view of Lemma 2.1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M174">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M175">View MathML</a>

(2.10)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M177">View MathML</a>.

By (2.9) and (2.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M178">View MathML</a>

so f is also Lipschitz on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M179">View MathML</a>.

For any ζ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M180">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M181">View MathML</a>

(2.11)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> by Fatou’s lemma, and it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M184">View MathML</a>

(2.12)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M187">View MathML</a>. Then we conclude

(2.13)

by (2.12) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> and (2.13) remains true if we restrict <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192">View MathML</a>, which is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113">View MathML</a> by Definition 2.4. The bounded linear functional ζ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> restricted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192">View MathML</a> is also a bounded linear functional, and we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M196">View MathML</a> to denote the functional restricted on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192">View MathML</a>.

We interpret (2.13) by saying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M196">View MathML</a> belongs to the subgradient at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M199">View MathML</a> of the convex functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M200">View MathML</a>

(2.14)

which is defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M202">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M187">View MathML</a>. In view of condition (A) and (2.12), we have

(2.15)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M208">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110">View MathML</a>.

Now we can apply Clarke’s abstract framework to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210">View MathML</a> with the following cast of characters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M211">View MathML</a> with the Lebesgue measure, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M212">View MathML</a>, which is a separable Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M213">View MathML</a>;

• let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M214">View MathML</a>, which is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M113">View MathML</a> denotes the space of measure essentially bounded functions mapping T to Y, equipped with the usual supremum norm by Definition 2.5;

• define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210">View MathML</a> on Z by (2.14);

• the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M218">View MathML</a> is measurable for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M186">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M110">View MathML</a> (see [20]), and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M221">View MathML</a> is a point at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M210">View MathML</a> is defined (finitely);

• the condition (2.6) in Clarke’s abstract framework is satisfied by (2.15).

By (2.12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M223">View MathML</a>

thus, every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M224">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M225">View MathML</a>

(2.16)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M226">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M228">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M230">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> by Definition 2.4. So for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M234">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M235">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M236">View MathML</a>

(2.17)

Combining (2.16) and (2.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M237">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M234">View MathML</a>. Then we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M239">View MathML</a>

and this completes the proof. □

Remark 2.7 The interpretation of expression (2.8) is as follows: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M179">View MathML</a> is an element in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M242">View MathML</a>, we deduce the existence of a measurable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M243">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M244">View MathML</a>

(2.18)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M246">View MathML</a>

and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M182">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a>.

Define a functional ϕ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M250">View MathML</a>

(2.19)

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M251">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a>, then we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M253">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M255">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a>, it is easy to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M257">View MathML</a>.

Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M258">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>, then we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M260">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M163">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M262">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M263">View MathML</a>, and it is easy to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M264">View MathML</a>.

Lemma 2.3The corresponding functionals<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M265">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M266">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M268">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M269">View MathML</a>

whereFsatisfies the condition (A) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M270">View MathML</a>, then the functional defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M271">View MathML</a>

(2.20)

is Lipschitz on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M273">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M274">View MathML</a>

(2.21)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M275">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M276">View MathML</a>.

Proof By direct computation, it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M277">View MathML</a>

(2.22)

In view of Lemma 2.2 and Remark 2.7, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M278">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M279">View MathML</a>

(2.23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M276">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M281">View MathML</a>, (2.21) holds by (2.22) and (2.23), and this completes the proof. □

Making use of Property 2.1 and Definition 2.3, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M282">View MathML</a>, BVP (1.1) is equivalent to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M283">View MathML</a>

(2.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M284">View MathML</a>. Therefore, we seek a solution u of BVP (2.24), which corresponds to the solution u of BVP (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M285">View MathML</a>.

Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M286">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M287">View MathML</a>

(2.25)

then we are in a position to give the definition of the solution of BVP (2.24).

Definition 2.9 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M288">View MathML</a> is called a solution of BVP (2.24) if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M289">View MathML</a> is differentiable for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>.

(ii) u satisfies (2.24).

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M97">View MathML</a>, andφis defined by (2.20). If assumption (A) is satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a>is a solution of the corresponding Euler equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M293">View MathML</a>, thenuis a solution of BVP (2.24) which, of course, corresponds to the solution of BVP (1.1).

Proof By Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M294">View MathML</a>

(2.26)

where for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M296">View MathML</a>.

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M297">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M298">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M299">View MathML</a>

By the Fubini theorem and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M300">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M301">View MathML</a>

Hence, by (2.26) we have, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M303">View MathML</a>

(2.27)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M304">View MathML</a> denotes the Canonical basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M307">View MathML</a>

The theory of Fourier series and (2.27) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M308">View MathML</a>

a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M310">View MathML</a>. According to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M311">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M312">View MathML</a>

(2.28)

a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M310">View MathML</a>.

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M315">View MathML</a>, we shall identify the equivalence class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M289">View MathML</a> given by its continuous representant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M317">View MathML</a>

(2.29)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>.

Therefore, it follows from (2.28) and the classical result of the Lebesgue theory that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M319">View MathML</a> is the classical derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M320">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M109">View MathML</a> which means that (i) in Definition 2.9 is verified.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M323">View MathML</a>, it remains to show that u satisfies (2.24). In fact, according to (2.29), we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M324">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M69">View MathML</a>. □

Lemma 2.5 ([32])

LetXbe a real reflexive Banach space. If the functionalψ: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M327">View MathML</a>is weakly lower semi-continuous and coercive, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M328">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M329">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M330">View MathML</a>. Moreover, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M331">View MathML</a>.

Lemma 2.6 ([32])

LetXbe a real reflexive Banach space, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M332">View MathML</a>is a locally Lipschitz function. If there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M333">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M334">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M335">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M336">View MathML</a>

(2.30)

andψsatisfies the nonsmooth (P.S.) condition with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M337">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M338">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M339">View MathML</a>andcis a critical value ofψ.

Definition 2.10 ([37])

Assume that the compact group G acts diagonally on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M340">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M341">View MathML</a>

where V is a finite dimensional space. The action of G is admissible if every continuous equivariant map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M342">View MathML</a> has a zero, where U is an open bounded invariant neighborhood of 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M344">View MathML</a>.

Example 2.1 The antipodal action of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M345">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M346">View MathML</a> is admissible.

We consider the following situation:

(A1) The compact group G acts isometrically on the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M347">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M348">View MathML</a> is invariant and there exists a finite dimensional space V such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M349">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M350">View MathML</a> and the action of G on V is admissible.

Lemma 2.7 ([27])

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M351">View MathML</a>is an invariant locally Lipschitz functional. If, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M352">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M353">View MathML</a>such that

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M354">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M355">View MathML</a>;

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M356">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M358">View MathML</a>;

(A4) φsatisfies the nonsmooth (P.S.)ccondition for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M359">View MathML</a>.

Thenφhas an unbounded sequence of critical values.

Remark 2.8 The condition (A1) is needed for the proof of Lemma 2.7, see details in [27] and the references therein.

3 Main results and proofs of the theorems

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M360">View MathML</a>andFsatisfy the condition (A), and suppose the following conditions hold:

(B1) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M361">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M362">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M363">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M365">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305">View MathML</a>;

(B2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M367">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M368">View MathML</a>

uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>;

(B3) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M370">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M371">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M372">View MathML</a>

uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>.

Then system (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M375">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M376">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M377">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378">View MathML</a>. First, we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379">View MathML</a> is a bounded sequence. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M380">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M381">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M382">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M383">View MathML</a>

(3.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M295">View MathML</a>. It follows from (3.1) that

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M386">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M387">View MathML</a>.

By (A) and the nonsmooth (P.S.) condition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M388">View MathML</a>

is bounded, which combined with (3.2) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M361">View MathML</a>.

By Proposition 2.3, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379">View MathML</a> has a subsequence, also denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M394">View MathML</a>

(3.3)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M395">View MathML</a> is bounded, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M396">View MathML</a> is a positive constant.

Therefore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M397">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M398">View MathML</a> is the function from the nonsmooth (P.S.) condition, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M399">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M400">View MathML</a>

(3.4)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378">View MathML</a>, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M402">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M404">View MathML</a>.

By (3.4) and (3.5), it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M405">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M406">View MathML</a>, and hence that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M407">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M379">View MathML</a> admits a convergent subsequence.

In view of (B3), there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M410">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M411">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M412">View MathML</a>

(3.6)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M414">View MathML</a>. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M174">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M416">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M417">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>. Therefore, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M419">View MathML</a>

(3.7)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M15">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M423">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M424">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M425">View MathML</a>

(3.8)

by (3.6), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M426">View MathML</a> is a positive constant. Then there exists a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M427">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M428">View MathML</a>.

By (B2), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M429">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M430">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M431">View MathML</a>

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M433">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M434">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M435">View MathML</a>. Then it follows from (2.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M436">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M438">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M439">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M98">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M438">View MathML</a>. This implies all the conditions in Lemma 2.6 are satisfied, so there exists a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M442">View MathML</a> for φ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M443">View MathML</a>, and this completes the proof. □

Theorem 3.2LetFsatisfy (A), (B1), (B3) and the following conditions:

(B4) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M444">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M445">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M446">View MathML</a>

uniformly for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>;

(B5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M448">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a>and allxin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M305">View MathML</a>.

Then system (1.1) has an infinite number of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M451">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>for every positive integerksuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M453">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>.

Proof The proof that the functional φ satisfies the nonsmooth (P.S.) condition is the same as that of Theorem 3.1, so we omit it. We only need to verify other conditions in Lemma 2.7.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> is a separable and reflexive Banach space, there exist (see [38]) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M457">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M458">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M459">View MathML</a> , denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M460">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M461">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M462">View MathML</a>

(3.9)

and it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M463">View MathML</a> defined by (3.9) is a norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M464">View MathML</a>. Since all the norms of a finite dimensional normed space are equivalent, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M465">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M466">View MathML</a>

(3.10)

In view of (B3), there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M467">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M396">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M469">View MathML</a>

(3.11)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M471">View MathML</a>. It follows from (3.10) and (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M472">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M474">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M370">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M476">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M477">View MathML</a>

(3.12)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M478">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M479">View MathML</a>

(3.13)

then we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M480">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>. In fact, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M482">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M483">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M352">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M486">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M487">View MathML</a>

(3.14)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> is reflexive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89">View MathML</a> has a weakly convergent subsequence, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M89">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M91">View MathML</a>. We claim <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M492">View MathML</a>. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M493">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M494">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M495">View MathML</a>, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M496">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M493">View MathML</a>, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M492">View MathML</a>.

By Proposition 2.3, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M499">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M501">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M502">View MathML</a>. So we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M503">View MathML</a> by (3.14). In view of (B4), there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M504">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M505">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M506">View MathML</a>

(3.15)

for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M508">View MathML</a>. We conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M509">View MathML</a>

by (3.15), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M510">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M511">View MathML</a>.

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M512">View MathML</a>, it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M513">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M514">View MathML</a>

(3.16)

that is, condition (A3) in Lemma 2.7 is satisfied. In view of (3.12), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M515">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M516">View MathML</a>

and this shows condition (A2) in Lemma 2.7 is satisfied.

We have proved the functional φ satisfies all the conditions of Lemma 2.7, then φ has an unbounded sequence of critical values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M517">View MathML</a> by Lemma 2.7; we only need to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M518">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M95">View MathML</a>.

In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M451">View MathML</a> is a critical point of the functional φ, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M521">View MathML</a>, by Lemma 2.3 and Remark 2.7, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M522">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M523">View MathML</a>. Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M524">View MathML</a>

(3.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M525">View MathML</a>

(3.18)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M526">View MathML</a>, it is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M527">View MathML</a>

by (3.14). This completes the proof of Theorem 3.2. □

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M7">View MathML</a>satisfy the condition (A) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M529">View MathML</a>. Then BVP (1.1) has at least one solution which minimizesφon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>.

Proof By (3.1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M531">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M532">View MathML</a> is defined in (2.9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M533">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M505">View MathML</a> are constants. Hence, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M535">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M536">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M537">View MathML</a>

According to the same arguments in [36], φ is weakly lower semi-continuous. By Lemma 2.5, the proof of Theorem 3.3 is completed. □

4 Example

In this section, we give two examples to illustrate our results.

Example 4.1 In BVP (1.1), let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M538">View MathML</a>

It is easy to verify all the conditions in Theorem 3.2, so BVP (1.1) has infinitely many solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M539">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M541">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M378">View MathML</a>.

Example 4.2 In BVP (1.1), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M543">View MathML</a>. It is easy to verify all the conditions in Theorem 3.3, so BVP (1.1) has at least one solution which minimizes φ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/82/mathml/M249">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgement

The authors thank the anonymous referees for valuable suggestions and useful hints from others.

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