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On the solvability of the boundary value problem without initial condition for Schrödinger systems in infinite cylinders

Nguyen Manh Hung1 and Nguyen Thi Lien2*

Author Affiliations

1 National Institute of Education Management, Hanoi, Vietnam

2 Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

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Boundary Value Problems 2013, 2013:156  doi:10.1186/1687-2770-2013-156

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/156


Received:29 January 2013
Accepted:16 June 2013
Published:1 July 2013

© 2013 Hung and Lien; licensee Springer

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

Abstract

In this paper, we deal with the boundary value problems without initial condition for Schrödinger systems in cylinders. We establish several results on the existence and uniqueness of solutions.

Keywords:
generalized solution; problems without initial condition; cylinders

Introduction

The initial boundary value for the Schrödinger equation in cylinders with base containing conical points was established in [1]. Such a problem for parabolic systems was studied in Sobolev spaces with weights [2]. The boundary value problem without initial condition for parabolic equation was investigated in [3].

In the present paper, we consider the boundary value problem without initial condition for Schrödinger systems in cylinders. Firstly, following the method in [1], we prove the existence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1">View MathML</a> of problems with initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M2">View MathML</a>. Then, by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M3">View MathML</a>, the solvability of a problem without initial condition is obtained.

This paper is organized as follows. In the first section, we state the problem. In Section 2, we present the results on the unique solvability of problems with initial condition for Schrödinger systems in cylinders. The well-posedness of the problem without initial condition is dealt with in Sections 3 and 4.

1 Setting the problem

Let Ω be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M5">View MathML</a>) with boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M6">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M7">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M9">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M10">View MathML</a>, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M11">View MathML</a> to refer to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M13">View MathML</a> to refer to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M14">View MathML</a>. For each multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M15">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M17">View MathML</a>.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M21">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M22">View MathML</a>.

Let us introduce some functional spaces (see [4]) used in this paper.

We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M23">View MathML</a> to be the space of s-dimensional vector functions defined in Ω with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M24">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M25">View MathML</a> the space consisting of all vector functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M26">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M27">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M28">View MathML</a> is the space of vector functions with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M29">View MathML</a>

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M30">View MathML</a>

Especially, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M31">View MathML</a>.

Denote by the completion of infinitely differentiable vector functions vanishing near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M33">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M34">View MathML</a> norm.

Now we introduce a differential operator of order 2m

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M36">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M37">View MathML</a> matrices with the bounded complex-valued components in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M39">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M40">View MathML</a> is the transposed conjugate matrix to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M36">View MathML</a>). Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M42">View MathML</a>

We assume further that the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M43">View MathML</a> is -elliptic uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M45">View MathML</a>, which means there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M46">View MathML</a> independent of t and u such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M47">View MathML</a>

(1.1)

for all and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M45">View MathML</a>, where is a subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M51">View MathML</a>, the dense subset of infinitely differentiable complex s-dimensional vector functions with compact support in Ω.

Now we consider the following problem in the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M11">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M53">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M54">View MathML</a>

(1.3)

where ν is the unit vector of outer normal to the surrounding surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M13">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M56">View MathML</a>, a complex vector-valued function is called a generalized solution of problem (1.2)-(1.3) if and only if, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M59">View MathML</a>

(1.4)

holds for all , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62">View MathML</a>.

2 The unique solvability of problems with initial condition

Firstly, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M63">View MathML</a>, we study the following problem in the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M64">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M65">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M66">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M67">View MathML</a>

(2.3)

where ν is the unit vector of outer normal to the surrounding surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M68">View MathML</a>.

The solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M69">View MathML</a> is surveyed in the generalized sense. That means is a generalized solution if and only if, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M72">View MathML</a>

(2.4)

for all , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M74">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62">View MathML</a>.

In [1], the unique solvability of problem (2.1)-(2.3) is studied in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M77">View MathML</a>. Now, by the same method, we consider that problem in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M79">View MathML</a>.

Theorem 2.1Assume that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M80">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M81">View MathML</a>.

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M83">View MathML</a>, there exists a uniquely generalized solutionof problem (2.1)-(2.3) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M85">View MathML</a>

(2.5)

whereCis a nonnegative constant independent ofh, v, andf.

Proof The uniqueness is proved in a similar way as in [1]. We omit the details here. Now we prove the existence by the Galerkin approximating method. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M86">View MathML</a> is an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M51">View MathML</a> which is orthonormal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M88">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M89">View MathML</a>, we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M90">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M91">View MathML</a> is the solution of the ordinary differential system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M92">View MathML</a>

(2.6)

So, multiplying both sides of (2.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M93">View MathML</a> and taking the sum with respect to k from 1 to N, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M94">View MathML</a>

Adding this equation to its complex conjugate, integrating with respect to t from h to T, and then integrating by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M95">View MathML</a>

(2.7)

Using (1.1) and the Cauchy inequality, we receive from (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M96">View MathML</a>

(2.8)

Using the Gronwall-Bellman inequality, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M97">View MathML</a>, from (2.8) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M98">View MathML</a>

Multiplying both sides of this equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M99">View MathML</a> and integrating with respect to T from h to ∞, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M100">View MathML</a>

(2.9)

We denote by I, II, III, IV the terms from the first, second, third, fourth, respectively, of the right-hand sides of (2.9). We will give estimations for these terms. Firstly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M102">View MathML</a>

Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M103">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M104">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M105">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M106">View MathML</a>. Next, the term III can be estimated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M107">View MathML</a>

The last term, IV, is equal to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M108">View MathML</a>

Combining the above estimations, we get from (2.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M109">View MathML</a>

(2.10)

where the constant C is independent of h, N. □

From this inequality, by standard weakly convergent arguments (see [1]), we can conclude that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M110">View MathML</a> possesses a subsequence convergent to a vector function , which is a generalized solution of problem (2.1)-(2.3). Moreover, it follows from (2.10) that (2.5) holds.

3 The uniqueness of generalized solution of problem (1.2)-(1.3)

Theorem 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M115">View MathML</a>, problem (1.2)-(1.3) has no more than one solution.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M117">View MathML</a> are two generalized solutions of problem (1.2)-(1.3), set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M118">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M120">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M121">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M122">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M125">View MathML</a>.

From the definition of generalized solution, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M126">View MathML</a>

(3.1)

Adding (3.1) to its complex conjugate, we discover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M127">View MathML</a>

which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M128">View MathML</a>

(3.2)

Using the assumption of Theorem 3.1 and the Cauchy inequality, the left-hand sides of (3.2) can be estimated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M129">View MathML</a>

From (1.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M130">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M131">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M132">View MathML</a>

(3.3)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M133">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M134">View MathML</a>

Replacing them into (3.3), noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M135">View MathML</a>, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M136">View MathML</a>

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M137">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M138">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M139">View MathML</a>

where the positive constant C depends only on μ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M140">View MathML</a>.

Using the Gronwall-Bellman inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M141">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M142">View MathML</a> almost everywhere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M143">View MathML</a>. Because of the uniqueness of the solution of the problem with initial condition (2.1)-(2.3), we imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M144">View MathML</a> almost everywhere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M145">View MathML</a>. □

4 The existence of generalized solution

The generalized solution of problem (1.2)-(1.3) can be approximated by a sequence of solutions of problems with initial condition (2.1)-(2.3).

It is known that there is a smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M146">View MathML</a> which is equal to 1 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M147">View MathML</a>, is equal to 0 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M148">View MathML</a> and assumes value in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M149">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M150">View MathML</a> (see [[5], Th. 5.5] for more details). Moreover, we can suppose that all derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M146">View MathML</a> are bounded. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M152">View MathML</a> be an integer. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M153">View MathML</a>, we then get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M154">View MathML</a>

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M156">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M157">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M158">View MathML</a>

(4.2)

where the constant C is independent of f, h.

Let us consider generalized solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M160">View MathML</a> of problems (2.1)-(2.3) in cylinders <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M162">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M163">View MathML</a> is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M165">View MathML</a> respectively. With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1">View MathML</a> can be understood in with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M170">View MathML</a>.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M171">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M172">View MathML</a> is the generalized solution of problem (2.1)-(2.3) in cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M162">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M163">View MathML</a> is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M175">View MathML</a>. According to (2.5),

Because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M177">View MathML</a>

Because of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M179">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M181">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180">View MathML</a>. Repeating this argument, we discover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M183">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M180">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M185">View MathML</a> is a Cauchy sequence and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M1">View MathML</a> is convergent to u in .

In conclusion, we have satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M189">View MathML</a>

(4.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62">View MathML</a>.

Because of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M196">View MathML</a>, (4.3) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M197">View MathML</a>

(4.4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M202">View MathML</a>, sending <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M203">View MathML</a>, (4.4) is written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M204">View MathML</a>

(4.5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M58">View MathML</a>, , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M61">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M62">View MathML</a>.

That means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M209">View MathML</a> is a generalized solution of problem (1.2)-(1.3). We obtain our main result.

Theorem 4.1Assume that:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M210">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M211">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M213">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M214">View MathML</a>.

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/156/mathml/M83">View MathML</a>, there exists a uniquely generalized solutionof problem (1.2)-(1.3) satisfying

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors studied, read and approved the final manuscript.

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.30.

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