Open Access Research

Well-posedness of delay parabolic equations with unbounded operators acting on delay terms

Allaberen Ashyralyev12 and Deniz Agirseven3*

Author Affiliations

1 Department of Mathematics, Fatih University, Istanbul, 34500, Turkey

2 Department of Mathematics, ITTU, Gerogly Street, Ashgabat, 74400, Turkmenistan

3 Department of Mathematics, Trakya University, Edirne, 22030, Turkey

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Boundary Value Problems 2014, 2014:126  doi:10.1186/1687-2770-2014-126


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/126


Received:31 March 2014
Accepted:8 May 2014
Published:20 May 2014

© 2014 Ashyralyev and Agirseven; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In the present paper, the well-posedness of the initial value problem for the delay differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M4">View MathML</a>) in an arbitrary Banach space E with the unbounded linear operators A and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5">View MathML</a> in E with dense domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6">View MathML</a> is studied. Two main theorems on well-posedness of this problem in fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.

MSC: 35G15.

Keywords:
delay parabolic equations; well-posedness; fractional spaces; coercive stability estimates

1 Introduction

The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [1-13] and the references therein) and insight has developed over the last three decades. The theory of stability and coercive stability of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations (see [14-19]). It is well known that various initial-boundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M8">View MathML</a>

(1)

in an arbitrary Banach space E with the unbounded linear operators A and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5">View MathML</a> in E with dense domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M6">View MathML</a>. Let A be a strongly positive operator, i.e.A is the generator of the analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>) of the linear bounded operators with exponentially decreasing norm when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M13">View MathML</a>. That means the following estimates hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M14">View MathML</a>

(2)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5">View MathML</a> be closed operators.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> is called a solution of the problem (1) if the following conditions are satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> is continuously differentiable on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M20">View MathML</a>. The derivative at the endpoint <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M21">View MathML</a> is understood as the appropriate unilateral derivative.

(ii) The element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M23">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M24">View MathML</a>, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M25">View MathML</a> is continuous on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M20">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> satisfies the equation and the initial condition (1).

A solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> of the initial value problem (1) is said to be coercive stable (well-posed) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M29">View MathML</a>

(3)

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M30">View MathML</a>. We are interested in studying the coercive stability of solutions of the initial value problem under the assumption that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M31">View MathML</a>

(4)

holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. We have not been able to obtain the estimate (3) in the arbitrary Banach space E. Nevertheless, we can establish the analog of estimates (3) where the space E is replaced by the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M34">View MathML</a>) under an assumption stronger than (4). The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained.

The present paper is organized as follows. Section 1 is introduction. In Section 2, two main theorems on well-posedness of the initial value problem (1) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of the initial-boundary value problem for delay parabolic equations are obtained. Finally, Section 4 is our conclusion.

2 Theorems on well-posedness

The strongly positive operator A defines the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M35">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M34">View MathML</a>) consisting of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M37">View MathML</a> for which the following norms are finite:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M38">View MathML</a>

We consider the initial value problem (1) for delay differential equations of parabolic type in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M39">View MathML</a> of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M18">View MathML</a> defined on the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M41">View MathML</a> with values in a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a>. First, we consider the problem (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M5">View MathML</a> commute, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M45">View MathML</a>

(5)

Theorem 2.1Assume that the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M46">View MathML</a>

(6)

holds for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>, whereMis the constant from (2). Then for everyt, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M49">View MathML</a> , we have the following coercive stability estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M50">View MathML</a>

(7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51">View MathML</a>does not depend on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M53">View MathML</a>. Here, we put<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M54">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M55">View MathML</a>.

Proof It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M56">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M57">View MathML</a> is the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M58">View MathML</a>

(9)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M59">View MathML</a> is the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M60">View MathML</a>

(10)

First, we consider the problem (9). Using the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M61">View MathML</a>

(11)

the semigroup property, condition (5), and the estimates (2), (6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M62">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M65">View MathML</a>

(12)

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a>. Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M67">View MathML</a>

is true for t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> for some n. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M70">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M71">View MathML</a>

Using the estimate (12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M72">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M76">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> . Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M79">View MathML</a>

(13)

is true for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. Applying (9), the triangle inequality, condition (5), and the estimates (6) and (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M81">View MathML</a>

(14)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. Second, we consider the problem (10). To prove the theorem it suffices to establish the following stability inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M83">View MathML</a>

(15)

for the solution of the problem (10) for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M85">View MathML</a> . Using the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M86">View MathML</a>

(16)

the semigroup property, and the definition of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M88">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M91">View MathML</a>

(17)

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a>. Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M49">View MathML</a> , for some n. Using the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M95">View MathML</a>

(18)

the semigroup property, the definition of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87">View MathML</a>, the estimate (2), and condition (6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M97">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M101">View MathML</a>

(19)

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> . Applying equation (9), the triangle inequality, and condition (5) and estimates (6) and (19), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M104">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. This result completes the proof of Theorem 2.1. □

Now, we consider the problem (1) when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M106">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. Note that A is a strongly positive operator in a Banach spaces E iff its spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M108">View MathML</a> lies in the interior of the sector of angle φ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M109">View MathML</a>, symmetric with respect to the real axis, and if on the edges of this sector, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M111">View MathML</a> and outside it the resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M112">View MathML</a> is the subject to the bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M113">View MathML</a>

(20)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M114">View MathML</a>. First of all let us give lemmas from the paper [18] that will be needed in the sequel.

Lemma 2.1For anyzon the edges of the sector,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M115">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M116">View MathML</a>

and outside it the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M117">View MathML</a>

holds for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M118">View MathML</a>. Here and in the futureMand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M119">View MathML</a>are the same constants of the estimates (2) and (20).

Lemma 2.2Let for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M120">View MathML</a>the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M121">View MathML</a>with domain which coincide with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M23">View MathML</a>admit a closure<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M123">View MathML</a>bounded inE. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M124">View MathML</a>the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M125">View MathML</a>

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M126">View MathML</a>.

Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M127">View MathML</a>

(21)

holds for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. Here and in the futureεis some constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M129">View MathML</a>.

The application of Lemmas 2.1 and 2.2 enables us to establish the following fact.

Theorem 2.2Assume that the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M130">View MathML</a>

(22)

holds for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>the coercive stability estimate (7) holds.

Proof In a similar manner as in the proof of Theorem 2.1 we establish estimates for the solution of the problems (9) and (10), separately. First, we consider the problem (9). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. Then using (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M135">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M136">View MathML</a>

Using the estimates (2), (20), and condition (22), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M137">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. Now let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M140">View MathML</a>. By Lemma 2.1 and using the estimate (21), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M141">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. Using the triangle inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M144">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M147">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a>. In a similar manner as with Theorem 2.1 applying mathematical induction, one can easily show that it is true for every t. Therefore, to prove the theorem it suffices to establish the coercive stability inequality (15) for the solution of the problem (10). Now, we consider the problem (10). Exactly in the same manner, using (16), the semigroup property, and the definition of the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M87">View MathML</a>, we obtain (15) for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M63">View MathML</a>. Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M85">View MathML</a> for some n. Using (18) and the semigroup property, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M153">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M154">View MathML</a>

Using the estimate (2) and condition (22), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M155">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M157">View MathML</a> , and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. Now let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M140">View MathML</a>. By Lemma 2.2 and using the estimate (2) and condition (21), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M160">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. Using the triangle inequality and estimates for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M165">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M166">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> and λ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M64">View MathML</a>. This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M170">View MathML</a>

for every t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M69">View MathML</a> . This result completes the proof of Theorem 2.2. □

Note that these abstract results are applicable to the study of stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study the structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> for space operators in Banach spaces. The structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> for some space differential and difference operators in Banach spaces has been investigated (see [20-30]). In Section 3, applications of Theorem 2.1 to the study of the coercive stability of initial-boundary value problem for delay parabolic equations are given.

3 Applications

First, we consider the initial-boundary value problem for one dimensional delay differential equations of parabolic type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M175">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16">View MathML</a> is a sufficiently large number. We will assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M181">View MathML</a>. The problem (23) has a unique smooth solution. This allows us to reduce the initial-boundary value problem (23) to the initial value problem (1) in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M182">View MathML</a> with a differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183">View MathML</a> defined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M184">View MathML</a>

(24)

with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M185">View MathML</a>. Let us give a number of corollaries of the abstract Theorem 2.1.

Theorem 3.1Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M186">View MathML</a>

(25)

Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>the solutions of the initial-boundary value problem (23) satisfy the following coercive stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M188">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51">View MathML</a>is not dependent on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M192">View MathML</a>is the space of functions satisfying a Hölder condition with the indicator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M193">View MathML</a>.

The proof of Theorem 3.1 is based on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M194">View MathML</a>

and on the abstract Theorem 2.1, on the strong positivity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M196">View MathML</a> (see [31,32]) and on Theorem 3.2 on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M197">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M198">View MathML</a>.

Theorem 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M199">View MathML</a>, the norms of the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M200">View MathML</a>and the Hölder space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M201">View MathML</a>are equivalent[21].

Second, we consider the initial nonlocal boundary value problem for one dimensional delay differential equations of parabolic type,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M202">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16">View MathML</a> is a sufficiently large number. We will assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M181">View MathML</a>. The problem (26) has a unique smooth solution. This allows us to reduce the initial-boundary value problem (26) to the initial value problem (1) in Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M182">View MathML</a> with a differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183">View MathML</a> defined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M211">View MathML</a>

(27)

with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M212">View MathML</a>. Let us give a number of corollaries of the abstract Theorem 2.1.

Theorem 3.3Assume that condition (25) holds. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>the solutions of the initial-boundary value problem (26) satisfy the following coercive stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M214">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M51">View MathML</a>is not dependent on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a>.

The proof of Theorem 3.3 is based on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M218">View MathML</a>

and on the abstract Theorem 2.1, on the strong positivity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M196">View MathML</a> (see [6]) and on Theorem 3.4 on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M221">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M198">View MathML</a>.

Theorem 3.4For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M223">View MathML</a>, the norms of the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M200">View MathML</a>and the Hölder space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M201">View MathML</a>are equivalent[6].

Third, we consider the initial value problem on the range

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M226">View MathML</a>

for 2mth order multidimensional delay differential equations of parabolic type,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M227">View MathML</a>

(28)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a> are sufficiently smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M16">View MathML</a> is a sufficiently large number. We will assume that the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M233">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M234">View MathML</a>

of the differential operator of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M235">View MathML</a>

(29)

acting on functions defined on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M236">View MathML</a>, satisfies the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M237">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M238">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M239">View MathML</a>. The problem (28) has a unique smooth solution. This allows us to reduce the initial value problem (28) to the initial value problem (1) in Banach space E with a strongly positive operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M240">View MathML</a> defined by (29). Let us give a number of corollaries of the abstract Theorem 2.1.

Theorem 3.5Assume that condition (25) holds. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M2">View MathML</a>the solutions of the initial-boundary value problem (28) satisfy the following coercive stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M242">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M243">View MathML</a>does not depend on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M178">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M179">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M246">View MathML</a>is the space of functions satisfying a Hölder condition with the indicator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M247">View MathML</a>.

The proof of Theorem 3.5 is based on the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M248">View MathML</a>

and on the abstract Theorem 2.1, on the strong positivity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M183">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M250">View MathML</a>, and on the equivalence of the norms in the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M252">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M253">View MathML</a>[20,23].

4 Conclusion

In the present paper, two theorems on the well-posedness of the initial value problem for the delay parabolic differential equations with unbounded operators acting on delay terms in fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/126/mathml/M7">View MathML</a> are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

This work is supported by Trakya University Scientific Research Projects Unit (Project No: 2010-91).

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