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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Elliptic problems with nonhomogeneous boundary condition and derivatives of nonlinear terms

Dumitru Motreanu1 and Viorica V Motreanu2*

Author Affiliations

1 Département de Mathématiques, Université de Perpignan, Perpignan, 66860, France

2 Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, 84105, Israel

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Boundary Value Problems 2014, 2014:6  doi:10.1186/1687-2770-2014-6


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/6


Received:22 October 2013
Accepted:6 December 2013
Published:7 January 2014

© 2014 Motreanu and Motreanu; 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

The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.

MSC: 35H30, 35A16.

Keywords:
quasilinear elliptic problem; nonhomogeneous Dirichlet boundary conditions; existence result

1 Introduction

The aim of the paper is two-fold: first, to study nonlinear elliptic problems under nonhomogeneous Dirichlet boundary condition; second, to incorporate in the problem statement nonlinearities exhibiting derivatives of the solution. These requirements need to develop a nonstandard approach, in particular prevent the use of variational methods.

Specifically, we study two problems on a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M2">View MathML</a>) with Lipschitz boundary Ω. We first consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M3">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M6">View MathML</a> are Carathéodory functions (that is, they are measurable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7">View MathML</a> and continuous in the other variables), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M8">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M9">View MathML</a> denotes the space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M10">View MathML</a> sized symmetric matrices. In the following definition we make clear what we understand by solution to problem (1).

Definition 1 A (weak) solution of problem (1) is an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M15">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M16">View MathML</a>

Next we focus on nonhomogeneous Dirichlet problems where, contrary to problem (1), the dependence with respect to the gradient ∇u of the solution u is not expressed in a divergence form, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M17">View MathML</a>

(2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M6">View MathML</a> and g are as in problem (1), while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M21">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22">View MathML</a>) are Carathéodory functions. The meaning of solution of problem (2) is as follows.

Definition 2 A (weak) solution of problem (2) is an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M15">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M30">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M31">View MathML</a>.

Problems of type (1) and (2) have been investigated in settings that are different from ours (see, e.g., [1-6]). For instance, problem (1) is studied in [3] when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M33">View MathML</a> with functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M35">View MathML</a> corresponding to certain physical models, as described by Reynolds equation where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M36">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M39">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M40">View MathML</a>. Whereas many of the previous results on problems (1) and (2) involve technical and somewhat restrictive assumptions on the data, the purpose of the present paper is to provide an elementary resolution of problems (1) and (2) in geometrically relevant situation. As an example of such a geometrically relevant situation, we mention the assumption on the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M41">View MathML</a> in problem (1) to vanish at two points.

Our results are stated as Theorems 1 and 2. They are existence and location theorems on problems (1) and (2), respectively, guaranteeing solutions in the sense of Definitions 1 and 2 that fulfill an estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42">View MathML</a> with given constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M43">View MathML</a>. This a priori estimate of the solution is derived through natural geometric hypotheses that can be directly checked. It is also worthwhile to remark that we cannot drop by translation the nonhomogeneous boundary conditions to become homogeneous because our hypotheses would be no longer verified. The arguments used in the proof are based on truncation techniques and Schauder’s fixed point theorem. We emphasize that, due to the type of assumptions we impose, it is essential in our approach to keep separate the two terms in divergence form appearing in the statement of (1) and (2). A careful inspection of our proofs shows that we rely on the linearity with respect to the gradient ∇u in the first divergence term and on the vanishing at suitable points in the second divergence term.

The rest of the paper is organized as follows. Section 2 is devoted to problem (1). Section 3 studies problem (2).

2 Result on problem (1)

Throughout the paper the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M45">View MathML</a> stands for the usual norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M47">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>, respectively. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M49">View MathML</a> we denote the Euclidean norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M50">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M51">View MathML</a> be the first eigenvalue of the negative Laplacian differential operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M52">View MathML</a>, which is known to be positive and characterized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M53">View MathML</a>

(3)

We suppose the following hypotheses on the data a, b, f, and g in problem (1):

(H1) There is a Carathéodory function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M55">View MathML</a>

(H2) There are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M56">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M57">View MathML</a> on Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M58">View MathML</a>

(H3) The functions a, f are bounded on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M59">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M60">View MathML</a>

(H4) There is a Carathéodory function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M61">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M62">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M63">View MathML</a>

Remark 1 The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65">View MathML</a> are not solutions of problem (1), unless <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M66">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M67">View MathML</a> on Ω. Thus, in general, problem (1) has no evident solution.

Remark 2 Due to their different structure and requirements, the two terms in (1) that are in divergence form cannot be combined.

Remark 3 The last part of hypothesis (H4) incorporates the monotonicity condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M68">View MathML</a>

as well as the Lipschitz condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M69">View MathML</a>

and it is more general than both of them.

The result that we set forth in this section is the following theorem ensuring existence and location of solution for problem (1).

Theorem 1Assume that hypotheses (H1)-(H4) are satisfied. Then problem (1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11">View MathML</a>in the sense of Definition 1 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M71">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65">View MathML</a>given in (H2).

Proof Consider the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M74">View MathML</a>

which is a nonempty, bounded, closed, convex subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>.

Claim 1: Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M77">View MathML</a> of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M78">View MathML</a>

Note that Claim 1 is equivalent to solving uniquely the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M79">View MathML</a>

(4)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M81">View MathML</a> in (4) are expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M82">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M83">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M84">View MathML</a>. Notice that the operators A and B are well defined due to our hypotheses.

With the fixed element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, let us introduce the Carathéodory map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M86">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M87">View MathML</a>

From hypotheses (H1), (H3), (H4), and because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M89">View MathML</a> satisfies the properties: there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M90">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M91">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M92">View MathML</a>

(6)

Estimate (5) guarantees that the operator A is bounded (in the sense to be bounded on bounded sets). It is easily seen that (6) implies that A is coercive, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M93">View MathML</a>

Moreover, relations (5)-(6) ensure that the operator A is maximal monotone, so pseudomonotone (see, e.g., [[7], §2.3.1]). Since A is bounded, coercive, and pseudomonotone, it is surjective (see, e.g., [[7], Theorem 2.99]), whence the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94">View MathML</a> in Claim 1. The uniqueness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94">View MathML</a> is a direct consequence of (6) (notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M96">View MathML</a>). This establishes Claim 1.

Now, taking advantage of Claim 1, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M98">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94">View MathML</a> is the unique element corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> as proved in Claim 1.

Claim 2: The mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M104">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>. Using the definition of T and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M109">View MathML</a> as a test function in Claim 1 (written with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M110">View MathML</a> and u), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M111">View MathML</a>

Combining this formula with (H1), (H3), (H4), (3) and the Cauchy-Schwarz inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M112">View MathML</a>

Taking into account hypothesis (H4) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M113">View MathML</a>

(7)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M114">View MathML</a>

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M115">View MathML</a>

(8)

To this end, we show that any subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M116">View MathML</a> possesses a subsequence converging to 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>, we have that, along a relabeled subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M120">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7">View MathML</a>. Invoking (H3), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M122">View MathML</a>, with some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M123">View MathML</a>. Through Lebesgue’s dominated convergence theorem, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M124">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M125">View MathML</a>, so (8) holds true.

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M126">View MathML</a>

Then, in view of (7), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M127">View MathML</a>. Since the domain Ω is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M128">View MathML</a>, we can make use of the Poincaré inequality for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M129">View MathML</a>, which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M130">View MathML</a>, whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M131">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>. This establishes Claim 2.

With the truncation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M133">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M134">View MathML</a>

(9)

consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> introduced as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M136">View MathML</a>

(10)

Note that S takes values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M137">View MathML</a>.

Claim 3: The mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> has a fixed point.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97">View MathML</a> is continuous by Claim 2 (thus a fortiori<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M140">View MathML</a> is continuous) and τ is a bounded continuous function, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is continuous. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is a compact operator. To this end, it suffices to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Because of the compact embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>, it is sufficient to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>. By the definition of T and inserting therein the test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M151">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M152">View MathML</a>

Then, as in the proof of Claim 2, from assumptions (H1) and (H4) we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M153">View MathML</a>

whence, by (H4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M154">View MathML</a>

(11)

with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M155">View MathML</a> independent of u.

Using (11), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M156">View MathML</a>

with constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M157">View MathML</a> independent of u. It follows that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>, so according to what was said before, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is compact. Consequently, Schauder’s fixed point theorem can be applied (see, e.g., [[8], p.452]), through which it follows that S admits a fixed point in C. This shows Claim 3.

Claim 4: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> be a fixed point of S. Then there holds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M162">View MathML</a>.

The existence of a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164">View MathML</a> is ensured by Claim 3. Fix such a point u and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>. In order to deduce the desired conclusion from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164">View MathML</a>, it suffices to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M167">View MathML</a> a.e. in Ω. We only verify the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M168">View MathML</a> a.e. in Ω because the proof of the other inequality is similar. By virtue of hypothesis (H2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M169">View MathML</a> on Ω (in the sense of traces), hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170">View MathML</a> on Ω and so the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M171">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M52">View MathML</a> (see, e.g., [[7], p.35]). Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M173">View MathML</a> as a test function in the definition of T gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M174">View MathML</a>

which reads as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M175">View MathML</a>

(12)

By the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M164">View MathML</a> and from (10) we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M177">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M7">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M179">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180">View MathML</a>. Then hypothesis (H2) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M182">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180">View MathML</a>. Consequently, (12), (H1), and (H4) entail

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M184">View MathML</a>

whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M185">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M186">View MathML</a>. On the other hand, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M188">View MathML</a>. Altogether, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187">View MathML</a> in Ω. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M190">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170">View MathML</a> a.e. in Ω, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M192">View MathML</a> a.e. in Ω. This proves Claim 4.

By Claims 3 and 4, the operator T admits a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>. Then the definition of T implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M194">View MathML</a>, so u is a solution of problem (1). In addition, the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42">View MathML</a> a.e. in Ω. The proof of Theorem 1 is complete. □

3 Result on problem (2)

The hypotheses on the data a, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M197">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M198">View MathML</a>), f, and g in problem (2) that we suppose are as follows: (H1) in Section 2,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M57">View MathML</a> on Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M202">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>) There exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M205">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M206">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M208">View MathML</a> and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M22">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M211">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M212">View MathML</a>

Remark 4 As in the case of problem (1), we note that the constant functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M214">View MathML</a> are not solutions of problem (2), unless <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M66">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M216">View MathML</a> on Ω.

Now we state our result of existence and location of solutions for problem (2).

Theorem 2Assume that (H1), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>), and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>) are satisfied. Then problem (2) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M11">View MathML</a>in the sense of Definition 2 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M221">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65">View MathML</a>as in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>).

Proof We follow the pattern of proof of Theorem 1. Hence, using the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65">View MathML</a> prescribed in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>), we consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M228">View MathML</a>

which is a nonempty, bounded, closed, convex subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. We proceed by proving four claims regarding problem (2) that correspond to those in the proof of Theorem 1 for problem (1). We provide the proof since there are some differences with respect to the proof of Theorem 1.

Claim 1: For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M77">View MathML</a> of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M232">View MathML</a>

As in the proof of Theorem 1, first we note that Claim 1 is equivalent to proving that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M233">View MathML</a>

(13)

admits a unique solution, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M234">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M235">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, by the Cauchy-Schwarz inequalities in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a> and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M50">View MathML</a>, as well as (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>) and (3), we derive the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M240">View MathML</a>

(14)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M84">View MathML</a>. Using the Cauchy-Schwarz inequality in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>, the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>) and (14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M245">View MathML</a>

which ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M246">View MathML</a> is a continuous bilinear form. From (H1), the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>) and (14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M249">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M250">View MathML</a> (as postulated in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>)), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M246">View MathML</a> is also coercive.

On the basis of the reasoning in (14), the following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M253">View MathML</a>

(15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M31">View MathML</a>. Taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>), (15) and (3), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M257">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M258">View MathML</a> stands for the Lebesgue measure of Ω. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M259">View MathML</a> is linear and continuous. The properties of the mappings A and B permit to apply the Lax-Milgram theorem, through which we conclude that problem (13) admits a unique solution. This establishes Claim 1.

As in the proof of Theorem 1, we introduce the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M98">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M94">View MathML</a> given in Claim 1.

Claim 2: The mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M97">View MathML</a> is continuous.

In order to prove this assertion, we proceed as in the proof of Claim 2 in Theorem 1. Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> and consider a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M105">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M271">View MathML</a>

(16)

A straightforward calculation entails

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M272">View MathML</a>

(17)

Combining (H1), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>), (16), (17), (3), and the Cauchy-Schwarz inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M275">View MathML</a>

(18)

with μ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M276">View MathML</a> in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>).

Proceeding as in (8) we show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M278">View MathML</a>

(19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M279">View MathML</a>

(20)

Now it suffices to combine (18), (19), (20) and recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M250">View MathML</a> (see (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>)) to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M127">View MathML</a>. Then, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M128">View MathML</a> and Ω is bounded, by the Poincaré inequality, we also deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M130">View MathML</a>. This amounts to saying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M131">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>, which proves Claim 2.

Following the approach developed in the proof of Theorem 1, we introduce the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> given by (10), with the truncation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M133">View MathML</a> defined in (9) corresponding to the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M65">View MathML</a> in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>).

Claim 3: The mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> has a fixed point.

Claim 2 readily implies that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is continuous. Let us check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is a compact operator. To see this, it suffices to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Thanks to the compactness of the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>, this reduces to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>. To this end, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> and denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>. We can argue as in the proof of Theorem 1 by relying now on the present hypotheses. We obtain from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a> with the test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M151">View MathML</a>, in conjunction with (H1), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M203">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>), that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M307">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M90">View MathML</a> is a constant independent of u. In view of hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M310">View MathML</a>

(21)

with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M123">View MathML</a> independent of u. Using (21) and the definition of S, we get the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M312">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M155">View MathML</a> independent of u. We conclude that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M143">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M48">View MathML</a>, so relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M46">View MathML</a>. Therefore the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M135">View MathML</a> is compact. This enables us to apply Schauder’s fixed point theorem (see, e.g., [[8], p.452]), which implies that S possesses a fixed point in C. Claim 3 is thus shown.

Claim 4: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> is a fixed point of S, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M162">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> be a fixed point of S and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a>. In order to show that u is a fixed point of T, it is needed to be fulfilled <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M167">View MathML</a> a.e. in Ω. The proof is done following the pattern of the corresponding part in the proof of Theorem 1. We outline the proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M192">View MathML</a> a.e. in Ω (the proof of the other inequality is similar).

Testing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M108">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M325">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M326">View MathML</a>

(22)

In <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180">View MathML</a> it is true that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M328">View MathML</a>

Then hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M199">View MathML</a>) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M330">View MathML</a>

Combining with (22), (H1), and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M207">View MathML</a>) entails

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M332">View MathML</a>

It turns out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M185">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M180">View MathML</a>. Also, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M188">View MathML</a>. Consequently, the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M187">View MathML</a> in Ω is valid, which results in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M170">View MathML</a> a.e. in Ω because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M190">View MathML</a>. This reads as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M340">View MathML</a> a.e. in Ω, so Claim 4 is fulfilled.

Now we can conclude the proof. Claims 3 and 4 ensure that there exists a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a> of the operator T. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M194">View MathML</a> and u is a solution of problem (2). Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M76">View MathML</a>, we also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/6/mathml/M42">View MathML</a> a.e. in Ω. The desired conclusion is achieved. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

DM and VVM jointly worked and obtained all the results presented in the paper and participated equally in the preparation of the paper. Both authors read and approved the final manuscript.

Acknowledgements

The second author is supported by the Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement No. PIEF-GA-2010-274519).

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