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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

Finite-difference method for the hyperbolic system of equations with nonlocal boundary conditions

Allaberen Ashyralyev12* and Rahat Prenov2

Author Affiliations

1 Department of Mathematics, Fatih University, Istanbul, Turkey

2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan

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Advances in Difference Equations 2014, 2014:26  doi:10.1186/1687-1847-2014-26

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2014/1/26


Received:12 October 2013
Accepted:26 December 2013
Published:20 January 2014

© 2014 Ashyralyev and Prenov; licensee Springer.

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

Abstract

In the present paper, the finite-difference method for the initial-boundary value problem for a hyperbolic system of equations with nonlocal boundary conditions is studied. The positivity of the difference analogy of the space operator generated by this problem in the space C with maximum norm is established. The structure of the interpolation spaces generated by this difference operator is investigated. The positivity of this difference operator in Hölder spaces is established. In applications, stability estimates for the solution of the difference scheme for a hyperbolic system of equations with nonlocal boundary conditions are obtained. A numerical example is applied.

MSC: 35L40, 35L45.

Keywords:
hyperbolic system of equations; nonlocal boundary value problems; difference schemes; interpolation spaces; positivity of the difference operator; stability estimates

1 Introduction

Nonlocal problems are widely used for mathematical modeling of various processes of physics, ecology, chemistry, and industry, when it is impossible to determine the boundary or initial values of the unknown function. The method of operators as a tool for the investigation of the solution of local and nonlocal problems for partial differential equations in Hilbert and Banach spaces has been systematically developed by several authors (see, e.g., [1-27]). It is well known that (see, e.g., [28-32] and the references given therein) many application problems in fluid mechanics, physics, mathematical biology, and chemistry were formulated as nonlocal mathematical models. Note that such problems were not well studied in general.

In the paper [33], the initial-boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M1">View MathML</a>

(1)

for the hyperbolic system of equations with nonlocal boundary conditions was considered. Here

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M2">View MathML</a>

(2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M3">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M4">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M5">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M7">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M8">View MathML</a>) are given smooth functions and they satisfy all compatibility conditions which guarantee the problem (1) has a smooth solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M9">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M10">View MathML</a>. As noted in the paper [33], the problem of sound waves [34] and the problem of the expansion of electricity oscillations [35] can be replaced by the problem (1). Note that, we have the nonclassical initial-boundary value problem (1) with boundary conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M12">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M13">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M14">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M15">View MathML</a>. These conditions are given on two boundary points. It is clear that it is impossible to determine the boundary values of the unknown function. So, these conditions are not local.

Let E be a Banach space and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M16">View MathML</a> be a linear unbounded operator densely defined in E. We call A a positive operator in the Banach space if the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M17">View MathML</a> has a bounded inverse in E for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a>, and the following estimate holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M19">View MathML</a>

(3)

Throughout the present paper, M is defined as a positive constant. However, we will use <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M20">View MathML</a> to stress the fact that the constant depends only on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M21">View MathML</a> .

For a positive operator A in the Banach space E, let us introduce the fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M22">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M23">View MathML</a>) consisting of those <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M24">View MathML</a> for which the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M25">View MathML</a>

is finite.

Let us introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M26">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M27">View MathML</a>) of all continuous vector functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M28">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M29">View MathML</a> and satisfying a Hölder condition for which the following norm is finite:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M30">View MathML</a>

Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M31">View MathML</a> is the Banach space of all continuous vector functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M28">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M29">View MathML</a> with norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M34">View MathML</a>

We consider the space operator A generated by the problem (1) defined by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M35">View MathML</a>

(4)

with domain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M36">View MathML</a>

The Green’s matrix function of A was constructed. The positivity of the operator A in the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37">View MathML</a> was established. It was proved that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38">View MathML</a> the norms in spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M39">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M40">View MathML</a> are equivalent. The positivity of A in the Hölder spaces of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M40">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38">View MathML</a> was proved. In applications, stability estimates for the solution of the problem (1) for the hyperbolic system of equations with nonlocal boundary conditions were obtained.

In the present paper, the finite-difference method for the initial value problem for the hyperbolic system of equations with nonlocal boundary conditions is applied. The positivity of the difference analogy of the space operator A defined by equation (1) in the difference analogy of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37">View MathML</a> spaces is established. The structure interpolation spaces generated by this difference operator is studied. The positivity of this difference operator in Hölder spaces is established. In practice, stability estimates for the solution of the difference scheme for the hyperbolic system of equations with nonlocal boundary conditions are obtained. The method is illustrated by numerical example.

The organization of the present paper as follows. Section 1 is an introduction where we provide the history and formulation of the problem. In Section 2, the Green’s matrix function of the difference space operator is presented and positivity of this operator in the difference analogy of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M37">View MathML</a> spaces is proved. In Section 3, the structure of fractional spaces generated by this difference operator is investigated and positivity of this difference operator in Hölder spaces is established. In Section 4, stable difference schemes for the approximate solution of the problem (1) are constructed. A theorem on the stability for the first order of accuracy in the t difference scheme is proved. In Section 5, a numerical application is given. Finally, Section 6 is for our conclusion.

2 The Green’s matrix function of difference space operator and positivity

Let us introduce the Banach spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M45">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M46">View MathML</a>) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M47">View MathML</a> of all mesh vector functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M48">View MathML</a> defined on

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M49">View MathML</a>

with the following norms:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M50">View MathML</a>

We consider the difference space operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> generated by the problem (1) defined by the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M52">View MathML</a>

(5)

acting on the space of mesh vector functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M48">View MathML</a> defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M54">View MathML</a>, satisfying the conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M55">View MathML</a>

Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M56">View MathML</a>. We will study the resolvent of the difference space operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M57">View MathML</a>, i.e.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M58">View MathML</a>

(6)

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M59">View MathML</a>

(7)

Lemma 2.1For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a>, equation (7) is uniquely solvable and the following formula holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M61">View MathML</a>

(8)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M62">View MathML</a>

Here

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M63">View MathML</a>

(9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M64">View MathML</a>

(10)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M65">View MathML</a>

(11)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M66">View MathML</a>

Proof Using the resolvent equation (7), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M67">View MathML</a>

From that follows the following recursive formula:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M68">View MathML</a>

Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M69">View MathML</a>

From this formula and the nonlocal boundary condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M70">View MathML</a> it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M71">View MathML</a>

Then,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M72">View MathML</a>

(12)

Using the resolvent equation (7), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M73">View MathML</a>

From that follows the system of recursion formulas

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M74">View MathML</a>

Hence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M75">View MathML</a>

From this formula and the nonlocal boundary condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M76">View MathML</a> it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M77">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M78">View MathML</a>

Applying equation (12), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M79">View MathML</a>

From the last two formulas it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M80">View MathML</a>

(13)

Lemma 2.1 is proved. □

Lemma 2.2The following pointwise estimates hold; see equation (7):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M81">View MathML</a>

(14)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M82">View MathML</a>

(15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M83">View MathML</a>

(16)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M84">View MathML</a>

(17)

Here<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M85">View MathML</a>.

Proof It is easy to see that the estimates of equations (14), (15), and (16) follow from the triangle inequality. Applying the triangle inequality, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M86">View MathML</a>

(18)

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M87">View MathML</a>. Then, using the estimates of equations (14), (15), (16), and inequality (18), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M88">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M89">View MathML</a>. Then, using the estimates of equations (14), (15), (16), and inequality (18), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M90">View MathML</a>

Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M91">View MathML</a>. Then, using the estimates of equations (14), (15), (16), and inequality (18), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M92">View MathML</a>

Lemma 2.2 is proved. □

Theorem 2.1The operator<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M93">View MathML</a>has a bounded inverse in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M94">View MathML</a>for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a>and the following estimate holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M96">View MathML</a>

(19)

Proof Using the formula equation (13) and the triangle inequality, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M97">View MathML</a>

for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M98">View MathML</a>. Using the estimate of equation (15), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M99">View MathML</a>

Using the estimate of equation (16), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M100">View MathML</a>

Using the estimate of equation (17), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M101">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M102">View MathML</a>

From this it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M103">View MathML</a>

Theorem 2.1 is proved. □

3 The structure of fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104">View MathML</a> and positivity of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> in Hölder spaces

Clearly, the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> and its resolvent <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M107">View MathML</a> commute. By the definition of the norm in the fractional space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M108">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M109">View MathML</a>

Thus, from Theorem 2.1 it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> is a positive operator in the fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104">View MathML</a>. Moreover, we have the following result.

Theorem 3.1For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M38">View MathML</a>, the norms of the spaces<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M113">View MathML</a>and the Hölder space<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M114">View MathML</a>are equivalent uniformly with respect toh. Here

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M115">View MathML</a>

(20)

Proof For any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a> we have the obvious equality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M117">View MathML</a>

By equation (8), we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M118">View MathML</a>

(21)

Applying equation (21) and the following obvious equalities:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M119">View MathML</a>

and using the nonlocal boundary conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M120">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M121">View MathML</a>

Using this formula, the triangle inequality and the definition of spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M123">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M124">View MathML</a>

Here

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M125">View MathML</a>

Using the estimates

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M126">View MathML</a>

and the estimates of equations (14), (15), (16), and (17), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M127">View MathML</a>

Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M128">View MathML</a>

for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a>. This means that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M130">View MathML</a>

Let us prove the opposite inequality. For any positive operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M132">View MathML</a>

From the relation and formula (21) it follows that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M133">View MathML</a>

Consequently,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M134">View MathML</a>

whence

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M135">View MathML</a>

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M136">View MathML</a>

Then for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M137">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M138">View MathML</a>

Now let us estimate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M139">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M140">View MathML</a>

Note that it suffices to consider the case when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M141">View MathML</a>. Applying the scheme of the paper [33] and using equations (8), (9), (10), (11), and the estimates of equations (14), (15), (16), and (17), we can establish the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M142">View MathML</a>

(22)

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M143">View MathML</a>. Applying the triangle inequality and the estimate of equation (22), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M144">View MathML</a>

Thus for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M137">View MathML</a> we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M146">View MathML</a>

This means that the following inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M147">View MathML</a>

Theorem 3.1 is proved. □

Since the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> is a positive operator in the fractional spaces <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M104">View MathML</a>, from the result of Theorem 3.1 it follows that it is also a positive operator in the Hölder space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M150">View MathML</a>. Namely, we have the following.

Theorem 3.2The operator<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M93">View MathML</a>has a bounded inverse in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M152">View MathML</a>uniformly with respect tohfor any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M18">View MathML</a>and the following estimate holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M154">View MathML</a>

4 Applications

In this section we consider the application of results of Sections 2 and 3. For a positive operator A in E the following result was established in papers [36,37].

Theorem 4.1LetAbe a positive operator inE. Then it obeys the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M155">View MathML</a>

(23)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M156">View MathML</a>does not depend onτandk. Here<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M157">View MathML</a>is the Padé approximation of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M158">View MathML</a>near<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M159">View MathML</a>.

For a numerical solution of the initial-boundary value problem (1) the following difference scheme is presented:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M160">View MathML</a>

(24)

We introduce the Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M161">View MathML</a> of all continuous abstract mesh vector functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M162">View MathML</a>

defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M163">View MathML</a> with values in E, equipped with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M164">View MathML</a>

Note that the problem (24) can be written in the form of the abstract Cauchy problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M165">View MathML</a>

(25)

in a Banach space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M166">View MathML</a> with a positive operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a> defined by (5). Here <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M168">View MathML</a> is the given abstract vector function defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M169">View MathML</a> with values in E, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M170">View MathML</a> is the element of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M171">View MathML</a>. It is well known that (see, for example [3]) the formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M172">View MathML</a>

(26)

gives a solution of the problem (25) in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M161">View MathML</a>.

Theorem 4.2For the solution of the problem (25) the stability inequality holds:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M174">View MathML</a>

The proof of Theorem 4.2 is based on the positivity of the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M51">View MathML</a>, equation (26) and the estimate of equation (23).

Applying the results of Theorems 2.1 and 4.2, we get the following theorem.

Theorem 4.3The solution of the problem (24) satisfies the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M176">View MathML</a>

Applying results of Theorems 3.2, 4.1, and 4.2, we get the following theorem.

Theorem 4.4Assume that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M177">View MathML</a>

Then the solution of the problem (24) satisfies the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M178">View MathML</a>

Finally, one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates. Therefore, our interest in the present paper is studying the difference scheme equation (24) by numerical experiments. Applying this difference scheme, the numerical method is proposed in the following section for the numerical solution of the hyperbolic system of equations with nonlocal boundary conditions. The method is illustrated by a numerical example.

5 Numerical results

For the numerical result, the initial value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M179">View MathML</a>

(27)

for the hyperbolic system of equations with nonlocal boundary conditions is considered. Applying the difference scheme equation (24), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M180">View MathML</a>

(28)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M181">View MathML</a>

We get the system of equations in the matrix form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M182">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M183">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M184">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M185">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M186">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M187">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M188">View MathML</a>

Thus, we have the first-order difference equation with respect to k matrix coefficients. To solve this difference equation we have the following procedure:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M189">View MathML</a>

(29)

For their comparison, the errors are computed by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M190">View MathML</a>

of the numerical solutions. The numerical solutions are recorded for different values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M191">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M192">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M193">View MathML</a> represent the numerical solutions of these difference schemes at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M194">View MathML</a>. The errors are given in Table 1 for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M195">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M196">View MathML</a>, respectively.

Table 1. Difference scheme

6 Conclusion

In the present study, the finite-difference method for the initial-boundary value problem for the hyperbolic system of equations with nonlocal boundary conditions is studied. The positivity of the difference analogy of the space operator generated by this problem in the space with maximum norm is established. The structure interpolation spaces generated by this difference operator is investigated. The positivity of this difference operator in Hölder spaces is established. In practice, stability estimates for the solution of the difference scheme for the hyperbolic system of equations with nonlocal boundary conditions are obtained. A numerical example is applied. Moreover, applying this approach we can construct the stable difference schemes for numerical solutions of the initial-boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2014/1/26/mathml/M197">View MathML</a>

(30)

for the hyperbolic system of semilinear equations with nonlocal boundary conditions. Of course, convergence estimates for the solution of these difference schemes can be obtained.

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

We would like to thank the reviewers whose careful reading, helpful suggestions, and valuable comments helped us to improve the manuscript.

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