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This article is part of the series Proceedings of 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013).

Open Access Research

Well-posedness of delay parabolic difference equations

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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Advances in Difference Equations 2014, 2014:18  doi:10.1186/1687-1847-2014-18


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2014/1/18


Received:8 November 2013
Accepted:16 December 2013
Published:16 January 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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The well-posedness of difference schemes of the initial value problem for delay differential equations with unbounded operators acting on delay terms in an arbitrary Banach space is studied. Theorems on the well-posedness of these difference schemes in fractional spaces are proved. In practice, the coercive stability estimates in Hölder norms for the solutions of difference schemes of the mixed problems for delay parabolic equations are obtained.

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

1 Introduction

Approximate solutions of the delay differential equations have been studied extensively in a series of works (see, for example, [1-6] and the references therein) and developed over the last three decades. In the literature mostly the sufficient condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M1">View MathML</a>

(1)

was considered for the stability of the following test delay differential equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M2">View MathML</a>

(2)

with the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M3">View MathML</a>

(3)

It is known that delay differential equations can be solved by applying standard numerical methods for ordinary differential equations without the presence of delay. However, it is difficult to generalize any numerical method to obtain a high order of accuracy algorithms, because high-order methods may not lead to efficient results. It is well known that even if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M6">View MathML</a> are arbitrary differentiable functions, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M7">View MathML</a> may not possess the higher-order derivatives for a sufficiently large t. Therefore, we have non-smooth solution of delay differential equations for given smooth data. This is the main difficulty in the study of the convergence of numerical methods for delay differential equations.

Delay partial differential equations arise from various applications, like in climate models, biology, medicine, control theory, and many others (see, for example, [7] and the references therein).

The theory of approximate solutions of delay partial differential equations has received less attention than delay ordinary differential equations. A situation which occurs in delay partial differential equations when the delay term is an operator of lower order with respect to the other operator term is widely investigated (see, for example, [7-9] and the references therein). In the case where the delay term is an operator of the same order with respect to other operator term, this is studied mainly in a Hilbert space (see, for example, [10] and the references therein). In fact there are very few papers where the delay term is an operator of the same order with respect to the other operator term, this being investigated in a general Banach space (see [11-14]) and in these works, the authors look only for partial differential equations under regular data. Additionally, approximate solutions of the delay parabolic equations in the case where the delay term is a simple operator of the same order with respect to the other operator term were studied recently in papers [15-19].

It is 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.advancesindifferenceequations.com/content/2014/1/18/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M8">View MathML</a>

(4)

in an arbitrary Banach space E with the unbounded linear operators A and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> in E with dense domains <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M10">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.advancesindifferenceequations.com/content/2014/1/18/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M11">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>) of the linear bounded operators with exponentially decreasing norm when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M13">View MathML</a>. That means the following estimates hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M14">View MathML</a>

(5)

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> be closed operators.

The strongly positive operator A defines the fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M18">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19">View MathML</a>) consisting of all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M20">View MathML</a> for which the following norms are finite:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M21">View MathML</a>

As noted in [19], it is important to study the stability of solutions of the initial value problem (4) for delay differential equations and of difference schemes for approximate solutions of problem (4) under the assumption that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M22">View MathML</a>

(6)

holds for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>. This assumption for the delay differential equation (2) follows from assumption (1) in the case when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M24">View MathML</a>. Unfortunately, we have not been able to obtain the stability estimate for the solution of problem (4) in the arbitrary Banach space E. Nevertheless, in [20], the coercive stability estimate for the solution of problem (4) was established, when the space E is replaced by the fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19">View MathML</a>) which is defined above under the condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M27">View MathML</a>

(7)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>, where M is the constant from equation (5). However, the condition (7) is stronger than (6) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M29">View MathML</a>. Finally, in papers [21,22], theorems on the well-posedness in Hölder spaces in t of the initial value problem for the delay parabolic equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M30">View MathML</a>

(8)

in an arbitrary Banach space E with the small positive parameter ε in the high derivative and with the unbounded linear operators A and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> in E with dense domains <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M10">View MathML</a> were established.

Additionally, using the first and second order of the accuracy implicit difference schemes for differential equations without the presence of delay, the first and second order of the accuracy implicit difference schemes,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M33">View MathML</a>

(9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M34">View MathML</a>

(10)

are presented for approximate solutions of the initial value problem (4). Here, we will put <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M35">View MathML</a>.

The main aim of present paper is to study the well-posedness of the difference schemes (9) and (10). We establish the coercive stability estimates in fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19">View MathML</a>) under the assumption (7). In practice, the coercive stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of the mixed problem of delay parabolic equations are obtained.

The paper is organized as follows. In Section 2, theorems on coercive stability of difference schemes (9) and (10) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of delay parabolic equations are obtained. Finally, Section 4 is our conclusion.

2 The well-posedness of difference schemes (9) and (10)

First, we consider the difference scheme (9) when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M38">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> commute, i.e.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M40">View MathML</a>

(11)

Theorem 1Assume that the condition (7) holds for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>, whereMis the constant from (5). Then for the solution of the difference scheme (9), the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M42">View MathML</a>

(12)

holds for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43">View MathML</a>. Here and in future we put<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M44">View MathML</a>if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M45">View MathML</a>.

Proof Let us consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. In this case

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M47">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M48">View MathML</a>

Let us estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M49">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M50">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43">View MathML</a>. Using the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M52">View MathML</a>

(13)

condition (11) and estimates (5) and (7), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M53">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M54">View MathML</a>

Making the substitution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M55">View MathML</a> and integrating by parts, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M56">View MathML</a>

Therefore <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M57">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M58">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M61">View MathML</a>

(14)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using formula (13), and the estimate (5), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M63">View MathML</a>

Applying the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M64">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M65">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M68">View MathML</a>

(15)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using the triangle inequality and the estimates (14) and (15), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M70">View MathML</a>

(16)

Applying mathematical induction, one can easily show that it is true for every k. Actually, suppose that the estimate (16) is true for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M72">View MathML</a> . Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M73">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M74">View MathML</a>

Using the estimate (16), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M75">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M76">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M72">View MathML</a> . Theorem 1 is proved. □

Now, we consider the difference scheme (9) when

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M78">View MathML</a>

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>.

Recall that (see, for example, [[23], Chapter 2, p.116]) A is a strongly positive operator in a Banach space E iff its spectrum <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M80">View MathML</a> lies in the interior of the sector of the angle φ, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M81">View MathML</a>, symmetric with respect to the real axis, and if on the edges of this sector, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M82">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M83">View MathML</a>, and outside it the resolvent <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M84">View MathML</a> is subject to the bound

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M85">View MathML</a>

(17)

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M86">View MathML</a>. First of all let us give lemmas from the paper [12] that will be needed in the sequel.

Lemma 1For anyzon the edges of the sector,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M87">View MathML</a>

and outside it the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M88">View MathML</a>

holds for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M89">View MathML</a>. Here and in the futureMand<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M90">View MathML</a>are the same constants as of the estimates (5) and (17).

Lemma 2Let for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M91">View MathML</a>the operator<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M92">View MathML</a>with domain which coincides with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M93">View MathML</a>permit the closure<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M94">View MathML</a>bounded inE. Then for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M95">View MathML</a>the following estimate holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M96">View MathML</a>

Here<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M97">View MathML</a>.

Suppose that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M98">View MathML</a>

(18)

holds for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>. Here and in the futureεis a constant, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M100">View MathML</a>.

The use of Lemmas 1 and 2 enables us to establish the following statement.

Theorem 2Assume that the condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M101">View MathML</a>

(19)

holds for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>. Then for the solution of the difference scheme (9), the coercive estimate (12) holds.

Proof Let us consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using formula (13), we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M104">View MathML</a>

Using the estimates (5), (17), and condition (19), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M105">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. Now let us estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M108">View MathML</a>. By Lemma 2 and using the estimate (20), we can obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M109">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. Using the triangle inequality, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M112">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M115">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using the triangle inequality and the last estimate and (15), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M117">View MathML</a>

In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 2 is proved. □

Now we consider the difference scheme (10). We have not been able to obtain the same result for the solution of the difference scheme (10) in spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> under assumption (7). Nevertheless, for the solution of the difference scheme (10) the coercive stability estimate in the norm of same fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M19">View MathML</a>) under the supplementary restriction of the operator A is established.

Theorem 3Suppose that the following estimates hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M121">View MathML</a>

(20)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M122">View MathML</a>

(21)

Then for the solution of the difference scheme (10), the coercive estimate (12) holds.

Proof Let us consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. In this case

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M124">View MathML</a>

(22)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M125">View MathML</a>

Let us estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M49">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M50">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43">View MathML</a>. Using formula (22), condition (11), and the estimates (5) and (7), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M129">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M132">View MathML</a>

(23)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using formula (22), the condition (21), and the estimate (5), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M134">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M137">View MathML</a>

(24)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using the triangle inequality and the estimates (23) and (24), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M139">View MathML</a>

(25)

In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 3 is established. □

Now, we consider the difference scheme (10) when

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M140">View MathML</a>

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a>. Suppose that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M142">View MathML</a>, with domain which coincides with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M93">View MathML</a>, permits the closure bounded in E and that the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M144">View MathML</a>

holds for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M12">View MathML</a> and some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M146">View MathML</a>.

Theorem 4Assume that all conditions of Theorem 2 and Theorem 3 are satisfied. Then for the solution of the difference scheme (10), the estimate (12) holds.

Proof Let us consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using formula (22), the estimates (5), (17), and the condition (19), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M148">View MathML</a>

Using the estimates (5), (17), and the condition (19), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M149">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. Now let us estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M152">View MathML</a>. By Lemma 2 and using the estimate (20), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M153">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. Using the triangle inequality, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M156">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M60">View MathML</a>. This shows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M159">View MathML</a>

(26)

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M46">View MathML</a>. Using the triangle inequality and the estimates (26) and (24), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M161">View MathML</a>

In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 4 is established. □

Note that these abstract results are applicable to the study of the coercive stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study structure of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> for space operators in Banach spaces. The structure of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M25">View MathML</a> for some space differential and difference operators in Banach spaces has been investigated in some papers [23-32]. In Section 3, applications of Theorem 1 to the study of the coercive stability of the difference schemes for delay parabolic equations are given.

3 Applications

First, the initial-boundary value problem for one-dimensional delay parabolic equations is considered:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M164">View MathML</a>

(27)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M165">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16">View MathML</a> is a sufficiently large number. It will be assumed that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M170">View MathML</a>. The discretization of problem (27) is carried out in two steps. In the first step, the uniform grid space

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M171">View MathML</a>

is defined. To formulate the result, one needs to introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M172">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M173">View MathML</a>) of the grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M174">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175">View MathML</a> satisfying the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M176">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M177">View MathML</a>

Here and in the future, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M178">View MathML</a> is the space of the grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M181">View MathML</a>

To the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182">View MathML</a> generated by problem (27), we assign the difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M184">View MathML</a>

acting in the space of grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179">View MathML</a> satisfying the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M176">View MathML</a>. With the help of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a>, we arrive at the initial-boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M188">View MathML</a>

(28)

for the system of ordinary differential equations. In the second step, problem (28) is replaced by the first-order accuracy in the difference scheme in t,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M189">View MathML</a>

(29)

Theorem 5Assume that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M190">View MathML</a>

(30)

Then for the solution of the difference scheme (29) the following coercive stability estimates hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M191">View MathML</a>

(31)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M193">View MathML</a>does not depend on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M194">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M195">View MathML</a>. Here and in the future we put

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M196">View MathML</a>

The proof of Theorem 5 is based on the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M197">View MathML</a>

and on the abstract Theorem 1, the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M199">View MathML</a>, and on the following theorem on the structure of the fractional space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M200">View MathML</a>.

Theorem 6For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M201">View MathML</a>the norms in the spaces<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M202">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M203">View MathML</a>are equivalent uniformly inh[25].

Second, the initial nonlocal boundary value problem for one-dimensional delay parabolic equations is considered:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M204">View MathML</a>

(32)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M165">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16">View MathML</a> is a sufficiently large number. It will be assumed that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M170">View MathML</a>. The discretization of problem (32) is carried out in two steps. In the first step, let us use the discretization in the space variable x. To formulate the result, one needs to introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M211">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M173">View MathML</a>) of the grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M175">View MathML</a> satisfying the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M215">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M216">View MathML</a> equipped with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M217">View MathML</a>

To the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182">View MathML</a> generated by problem (32) we assign the difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219">View MathML</a> by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M220">View MathML</a>

(33)

acting in the space of grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M179">View MathML</a> satisfying the conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M215">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M216">View MathML</a>. With the help of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219">View MathML</a>, we arrive at the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M225">View MathML</a>

(34)

for the system of ordinary differential equations. In the second step, problem (34) is replaced by the first-order accuracy of the difference scheme in t

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M226">View MathML</a>

(35)

Theorem 7Assume that all the conditions of Theorem 5 are satisfied. Then for the solution of the difference scheme (35) the coercive stability estimate (31) holds.

The proof of Theorem 7 is based on the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M227">View MathML</a>

and on the abstract Theorem 1, the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M219">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M229">View MathML</a>, and on the following theorem on the structure of the fractional space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M230">View MathML</a>.

Theorem 8For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M201">View MathML</a>the norms in the spaces<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M232">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M233">View MathML</a>are equivalent uniformly inh[27].

Third, the initial value problem on the range

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M234">View MathML</a>

for 2mth-order multidimensional delay differential equations of parabolic type is considered:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M235">View MathML</a>

(36)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M236">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M168">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M16">View MathML</a> is a sufficiently large number. We will assume that the symbol [<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M241">View MathML</a>]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M242">View MathML</a>

of the differential operator of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M243">View MathML</a>

(37)

acting on the functions defined on the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M244">View MathML</a>, satisfies the inequalities

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M245">View MathML</a>

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M246">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M247">View MathML</a>. The discretization of problem (36) is carried out in two steps. In the first step the uniform grid space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M249">View MathML</a>) is defined as the set of all points of the Euclidean space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M244">View MathML</a> whose coordinates are given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M251">View MathML</a>

The difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M252">View MathML</a> is assigned to the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M253">View MathML</a>, defined by equation (36). The operator

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M254">View MathML</a>

(38)

acts on functions defined on the entire space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248">View MathML</a>. Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M256">View MathML</a> is a vector with nonnegative integer coordinates,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M257">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M258">View MathML</a> is the unit vector of the axis <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M259">View MathML</a>.

An infinitely differentiable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M260">View MathML</a> of the continuous argument <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M261">View MathML</a> that is continuous and bounded together with all its derivatives is said to be smooth. We say that the difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> is a λth-order (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M263">View MathML</a>) approximation of the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182">View MathML</a> if the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M265">View MathML</a>

holds for any smooth function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M260">View MathML</a>. The coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M267">View MathML</a> are chosen in such a way that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> approximates in a specified way the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182">View MathML</a>. It will be assumed that the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> approximates the differential operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M182">View MathML</a> with any prescribed order [33,34].

The function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M272">View MathML</a> is obtained by replacing the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M273">View MathML</a> in the right-hand side of the equality (38) with the expression <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M274">View MathML</a>, respectively, and it is called the symbol of the difference operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M275">View MathML</a>.

It will be assumed that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M276">View MathML</a> and fixed x the symbol <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M277">View MathML</a> of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M278">View MathML</a> satisfies the inequalities

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M279">View MathML</a>

(39)

Suppose that the coefficient <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M267">View MathML</a> of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M281">View MathML</a> is bounded and satisfies the inequalities

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M282">View MathML</a>

(40)

With the help of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> we arrive at the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M284">View MathML</a>

(41)

for an infinite system of ordinary differential equations. Now, problem (41) is replaced by the first-order accuracy of the difference scheme in t

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M285">View MathML</a>

(42)

To formulate the result, one needs to introduce the spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M286">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M287">View MathML</a> of all bounded grid functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M288">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M248">View MathML</a>, equipped with the norms

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M290">View MathML</a>

Theorem 9Assume that all the conditions of Theorem 7 are satisfied. Then for the solution of the difference scheme (42) the following coercive stability estimates hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M291">View MathML</a>

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M43">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M293">View MathML</a>does not depend on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M194">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M195">View MathML</a>.

The proof of Theorem 9 is based on the estimate

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M296">View MathML</a>

and on the abstract Theorem 1, the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M183">View MathML</a> in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M298">View MathML</a>, and on the fact that the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M299">View MathML</a> norms are equivalent to the norms <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M300">View MathML</a> uniformly in h for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M301">View MathML</a> [[23], Chapter 4, p.283].

4 Conclusion

In the present paper, the well-posedness of the difference schemes for the approximate solutions of the initial value problem for delay parabolic equations with unbounded operators acting on delay terms in an arbitrary Banach space is established. Theorems on the coercive stability of these difference schemes in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the difference schemes for the approximate solutions of the mixed problems for delay parabolic equations are obtained. Note that in the present paper <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> is a time variable unbounded space operator acting on the delay term. The delay w is a positive constant. In general, it is interesting to consider the delay as a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M303">View MathML</a>, dependent on t. A well-known parabolic problem with delay used in population dynamics is the so-called Hutchinson equation where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> is a time variable bounded nonlinear space operator acting on the delay term [8,9]. It would be interesting to consider the case when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/18/mathml/M9">View MathML</a> is a nonlinear unbounded space operator acting on the delay term. Actually, it will be possible after establishing theorems on the existence, uniqueness, and stability of the solutions, and the smoothness property of the solutions, and obtaining a suitable contractivity condition of the numerical solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors read and approved the final version of the manuscript.

Acknowledgements

This work is supported by Trakya University Scientific Research Projects Unit (Project No: 2010-91). We would like to thank to the reviewers, whose careful reading, helpful suggestions, and valuable comments helped us to improve the manuscript.

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