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Study on integro-differential equation with generalized p-Laplacian operator

Li Wei1, Ravi P Agarwal23* and Patricia JY Wong4

Author affiliations

1 School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, China

2 Department of Mathematics, Texas A&M University — Kingsville, Kingsville, TX, 78363, USA

3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

4 School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

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Citation and License

Boundary Value Problems 2012, 2012:131  doi:10.1186/1687-2770-2012-131

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/131


Received:13 June 2012
Accepted:24 October 2012
Published:13 November 2012

© 2012 Wei et al.; licensee Springer

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

Abstract

We tackle the existence and uniqueness of the solution for a kind of integro-differential equations involving the generalized p-Laplacian operator with mixed boundary conditions. This is achieved by using some results on the ranges for maximal monotone operators and pseudo-monotone operators. The method used in this paper extends and complements some previous work.

MSC: 47H05, 47H09.

Keywords:
maximal monotone operator; pseudo-monotone operator; generalized p-Laplacian operator; integro-differential equation; mixed boundary conditions

1 Introduction

Nonlinear boundary value problems (BVPs) involving the p-Laplacian operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M1">View MathML</a> arise from a variety of physical phenomena such as non-Newtonian fluids, reaction-diffusion problems, petroleum extraction, flow through porous media, etc. Thus, the study of such problems and their generalizations have attracted numerous attention in recent years. Some of the BVPs studied in the literature include the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M2">View MathML</a>

(1.1)

whose existence results in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M3">View MathML</a> (for various ranges of p) can be found in [1-4]; a related BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M4">View MathML</a>

(1.2)

was tackled in [5-7] and later generalized to one that contains a perturbation term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M5">View MathML</a>[8,9]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M6">View MathML</a>

(1.3)

Motivated by Tolksdorf’s work [10] where the following Dirichlet BVP has been discussed:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M7">View MathML</a>

(1.4)

several generalizations have been investigated. These include [11-14]

(1.5)

(1.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M10">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M11">View MathML</a>, ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.

Inspired by all this research, recently we have studied the following nonlinear parabolic equation with mixed boundary conditions [15]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M12">View MathML</a>

(1.8)

We tackle the existence of solutions for (1.8) via the study of existence of solutions for two BVPs: (i) the elliptic equation with Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M13">View MathML</a>

(1.9)

and (ii) the elliptic equation with Neumann boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M14">View MathML</a>

(1.10)

By setting up the relations between the auxiliary equations (1.9) and (1.10) and by employing some results on ranges for maximal monotone operators, we showed that (1.8) has a unique solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M15">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M17">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M18">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M19">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M20">View MathML</a>.

In this paper, we shall employ the technique used in (1.8), viz. using the results on ranges for nonlinear operators, to study the existence and uniqueness of the solution to a nonlinear integro-differential equation with the generalized p-Laplacian operator. We note that most of the existing methods in the literature used to investigate such problems are based on the finite element method, hence our technique is new in tackling integro-differential equations. We shall consider the following nonlinear integro-differential problem with mixed boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M21">View MathML</a>

(1.11)

Our discussion is based on some results on the ranges for maximal monotone operators and pseudo-monotone operators in [16-18]. Some new methods of constructing appropriate mappings to achieve our goal are employed. Moreover, we weaken the restrictions on p and q. The paper is outlined as follows. In Section 2 we shall state the definitions and results needed, and in Section 3 we shall establish the existence and uniqueness of the solution to (1.11).

2 Preliminaries

Let X be a real Banach space with a strictly convex dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M23">View MathML</a> to denote the generalized duality pairing between X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>. For a subset C of X, we use IntC to denote the interior of C. We also use ‘→’ and ‘w-lim’ to denote strong and weak convergences, respectively.

Let X and Y be Banach spaces. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M25">View MathML</a> to denote that X is embedded continuously in Y.

The function Φ is called a proper convex function on X[17] if Φ is defined from X to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M26">View MathML</a>, Φ is not identically +∞ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M27">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M29">View MathML</a>.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M30">View MathML</a> is said to be lower-semicontinuous on X[17] if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M31">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M32">View MathML</a>.

Given a proper convex function Φ on X and a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M32">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M34">View MathML</a> the set of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M35">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M36">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M37">View MathML</a>. Such elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M38">View MathML</a> are called subgradients of Φ at x, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M39">View MathML</a> is called the subdifferential of Φ at x[17].

A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M40">View MathML</a> is said to be demi-continuous on X if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M41">View MathML</a> for any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M42">View MathML</a> strongly convergent to x in X. A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M40">View MathML</a> is said to be hemi-continuous on X if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M44">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M45">View MathML</a>[17].

With each multi-valued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M46">View MathML</a>, we associate the subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M47">View MathML</a> as follows [17]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M48">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M49">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a> is strictly convex, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M51">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M47">View MathML</a> is single-valued, which in this case is called the minimal section of A.

A multi-valued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M53">View MathML</a> is said to be monotone[18] if its graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M54">View MathML</a> is a monotone subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M55">View MathML</a> in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M56">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M58">View MathML</a>. The monotone operator B is said to be maximal monotone if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M54">View MathML</a> is not properly contained in any other monotone subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M55">View MathML</a>.

Definition 2.1[18]

Let C be a closed convex subset of X, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M61">View MathML</a>be a multi-valued mapping. Then A is said to be a pseudo-monotone operator provided that

(i) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M62">View MathML</a>, the image Ax is a nonempty closed and convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>;

(ii) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M42">View MathML</a> is a sequence in C converging weakly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M65">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M66">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M67">View MathML</a>, then to each element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M68">View MathML</a>, there corresponds an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M69">View MathML</a> with the property that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M70">View MathML</a>

(iii) for each finite-dimensional subspace F of X, the operator A is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M71">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a> in the weak topology.

Lemma 2.1[19]

Let Ω be a bounded conical domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M75">View MathML</a>; if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M77">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M78">View MathML</a>; if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M80">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M78">View MathML</a>.

Lemma 2.2[18]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M83">View MathML</a>is an everywhere defined, monotone, and hemi-continuous operator, thenBis maximal monotone. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M83">View MathML</a>is a maximal monotone operator such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M85">View MathML</a>, thenBis pseudo-monotone.

Lemma 2.3[18]

IfXis a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M86">View MathML</a>is a proper convex and lower-semicontinuous function, thenΦ is maximal monotone fromXto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>.

Lemma 2.4[18]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M89">View MathML</a>are two maximal monotone operators inXsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M90">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M91">View MathML</a>is maximal monotone.

Lemma 2.5[20]

LetXand its dual<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>be strictly convex Banach spaces. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M93">View MathML</a>is a closed linear operator and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M94">View MathML</a>is the conjugate operator ofS. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M95">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M96">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M97">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M98">View MathML</a>, thenSis a maximal monotone operator possessing a dense domain.

Lemma 2.6[18]

Any hemi-continuous mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M99">View MathML</a>is demi-continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M100">View MathML</a>.

Theorem 2.1[16]

LetXbe a real reflexive Banach space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M22">View MathML</a>being its dual space. LetCbe a nonempty closed convex subset ofX. Assume that

(i) the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M102">View MathML</a>is a maximal monotone operator;

(ii) the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M103">View MathML</a>is pseudo-monotone, bounded, and demi-continuous;

(iii) if the subsetCis unbounded, then the operatorBisA-coercive with respect to the fixed element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M104">View MathML</a>, i.e., there exists an element<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M105">View MathML</a>and a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M107">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M108">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M109">View MathML</a>.

Then the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M110">View MathML</a>has a solution.

3 Existence and uniqueness of the solution to (1.11)

We begin by stating some notations and assumptions used in this paper. Throughout, we shall assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M111">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M113">View MathML</a> be the dual space of V. The duality pairing between V and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M113">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M115">View MathML</a>. The norm in V will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M116">View MathML</a>, which is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M117">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M119">View MathML</a> be the dual space of W. The norm in W will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M120">View MathML</a>, which is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M121">View MathML</a>

In the integro-differential equation (1.11), Ω is a bounded conical domain of a Euclidean space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M123">View MathML</a>, Γ is the boundary of Ω with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M124">View MathML</a>[5], ϑ denotes the exterior normal derivative to Γ. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M126">View MathML</a> denote the Euclidean norm and the inner-product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M73">View MathML</a>, respectively. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M129">View MathML</a> is a given function, T and a are positive constants, and ε is a nonnegative constant. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M130">View MathML</a> is the subdifferential of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M132">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M133">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M134">View MathML</a> is a given function.

To tackle (1.11), we need the following assumptions which can be found in [5,14].

Assumption 1Green’s formula is available.

Assumption 2For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M136">View MathML</a>is a proper, convex, and lower-semicontinuous function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M137">View MathML</a>.

Assumption 3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M138">View MathML</a>and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M139">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M140">View MathML</a>is measurable for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M141">View MathML</a>.

We shall present a series of lemmas before we prove the main result.

Lemma 3.1Define the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M142">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M143">View MathML</a>

Then Φ is a proper, convex, and lower-semicontinuous mapping onV. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M144">View MathML</a>, the subdifferential of Φ, is maximal monotone.

Proof The proof of this lemma is analogous to that of Lemma 3.1 in [1]. We give the outline of the proof as follows.

Note that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M145">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M146">View MathML</a> is measurable, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M147">View MathML</a> denotes the minimal section of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M130">View MathML</a>. Since for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M149">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M150">View MathML</a>

it implies that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M151">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M152">View MathML</a> is measurable on Γ. Then from the property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131">View MathML</a>, we know that Φ is proper and convex on V.

To see that Φ is lower-semicontinuous on V, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M154">View MathML</a> in V. We may assume that there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M155">View MathML</a>, for simplicity, we still denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M155">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M157">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159">View MathML</a> a.e. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M160">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158">View MathML</a> and each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159">View MathML</a> a.e. since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M131">View MathML</a> is lower-semicontinuous for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M158">View MathML</a>. It then follows from Fatou’s lemma that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M159">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M166">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M167">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M168">View MathML</a> in V. This completes the proof. □

Lemma 3.2Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M169">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M170">View MathML</a>

ThenSis a linear maximal monotone operator possessing a dense domain inV.

Proof It is obvious that S is closed and linear.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M171">View MathML</a>, integrating by parts gives

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M173">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M174">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M175">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M176">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M177">View MathML</a>

Similarly, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M178">View MathML</a>,

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M180">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M181">View MathML</a>

In the same manner, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M182">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M183">View MathML</a>. Therefore, noting Lemma 2.5 the result follows. □

In view of Lemmas 2.3 and 2.4, we have the following result.

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M184">View MathML</a>is maximal monotone.

Lemma 3.4[14]

Define the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M185">View MathML</a>as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M186">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187">View MathML</a>is maximal monotone.

Lemma 3.5[14]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M188">View MathML</a>denote the closed subspace of all constant functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M189">View MathML</a>. LetXbe the quotient space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M190">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M191">View MathML</a>, define the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M192">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M193">View MathML</a>

Then, there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M194">View MathML</a>such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M196">View MathML</a>

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M197">View MathML</a>denotes the measure of Ω.

Definition 3.1 Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M198">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M199">View MathML</a>

Lemma 3.6The mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M200">View MathML</a>is everywhere defined, bounded, monotone, and hemi-continuous. Therefore, Lemma 2.2 implies that it is also pseudo-monotone.

Proof From Lemma 2.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M201">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M202">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M203">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M204">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M20">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M206">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M207">View MathML</a>. Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M209">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M210">View MathML</a> is a constant. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M211">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M212">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M213">View MathML</a>

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M214">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M215">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M217">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M218">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M219">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M220">View MathML</a>, we have

which implies that A is everywhere defined and bounded.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M222">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M220">View MathML</a>, we have

which also implies that A is everywhere defined and bounded.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187">View MathML</a> is monotone, we can easily see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M211">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M227">View MathML</a>

which implies that A is monotone.

To show that A is hemi-continuous, it suffices to show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M230">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M231">View MathML</a>. Noting the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M187">View MathML</a> is hemi-continuous and using the Lebesgue’s dominated convergence theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M233">View MathML</a>

Hence, A is hemi-continuous.

This completes the proof. □

Lemma 3.7The mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M200">View MathML</a>satisfies that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M235">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M236">View MathML</a>

(3.1)

as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M237">View MathML</a>inV.

Proof First, we shall show that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M238">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M239">View MathML</a>

is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M240">View MathML</a>

In fact, from Lemma 3.5, we know that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M151">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M242">View MathML</a>

where C is a positive constant. Thus,

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M244">View MathML</a>

(3.2)

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M245">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M246">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M247">View MathML</a>

(3.3)

In view of (3.2) and (3.3), we have shown that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M250">View MathML</a>.

Next, we shall show that A satisfies (3.1). In fact, we have

(3.4)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M222">View MathML</a>, then

(3.5)

From (3.2) and (3.3), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M254">View MathML</a>

Also,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M255">View MathML</a>

It follows from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M256">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249">View MathML</a>.

Moreover, we have

(3.6)

Therefore, it follows from (3.4), (3.5), and (3.6) that A satisfies (3.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M259">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M219">View MathML</a>, then

(3.7)

where M is a positive constant. We can easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M262">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249">View MathML</a>. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M264">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M265">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249">View MathML</a>; while if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M267">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M268">View MathML</a>

Hence, the right side of (3.7) tends to +∞ as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M249">View MathML</a>, which implies that A satisfies (3.1).

This completes the proof. □

Lemma 3.8If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M270">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M271">View MathML</a>a.e. on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M272">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M270">View MathML</a>, then from the definition of subdifferential, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M274">View MathML</a>

which implies that the result is true. □

We are now ready to prove the main result.

Theorem 3.1The integro-differential equation (1.11) has a unique solution inVfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M275">View MathML</a>.

Proof First, we shall show the existence of a solution. Noting Lemmas 2.6, 3.6, 3.7 and 3.3, and by using Theorem 2.1, we know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M276">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M277">View MathML</a>

(3.8)

Then we have for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M278">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M279">View MathML</a>

The definition of subdifferential implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M280">View MathML</a>

From the definition of S, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M281">View MathML</a>

(3.9)

Moreover,

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M283">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M284">View MathML</a>. Then we have

From the properties of a generalized function, we get

(3.11)

Noting (3.10) again, by using Green’s formula, we have

Then using (3.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M288">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M289">View MathML</a>.

In view of Lemma 3.8, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M290">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M272">View MathML</a>. Combining it with (3.8) and (3.11), we know that (1.11) has a solution in V.

Next, we shall prove the uniqueness of the solution. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M292">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M293">View MathML</a> be two solutions of (1.11). By (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M294">View MathML</a>

since S is monotone. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M295">View MathML</a> is monotone too, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M296">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/131/mathml/M297">View MathML</a>.

The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors approve the final manuscript.

Acknowledgements

Li Wei is supported by the National Natural Science Foundation of China (No. 11071053), the Natural Science Foundation of Hebei Province (No. A2010001482) and the Project of Science and Research of Hebei Education Department (the second round in 2010).

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