This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

Well-posedness of fractional parabolic equations

Allaberen Ashyralyev

Author Affiliations

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

Boundary Value Problems 2013, 2013:31  doi:10.1186/1687-2770-2013-31


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/31


Received:2 October 2012
Accepted:25 January 2013
Published:18 February 2013

© 2013 Ashyralyev; 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, we consider the abstract Cauchy problem for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M1">View MathML</a>

(1)

in an arbitrary Banach space E with the strongly positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>. The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of this problem is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2mth order multidimensional fractional parabolic equation are obtained.

MSC: 65M12, 65N12.

Keywords:
fractional parabolic equation; Basset problem; well-posedness; coercive stability

1 Introduction

It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, etc. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1-43] and the references given therein).

The role played by coercive stability inequalities (well-posedness) in the study of boundary value problems for parabolic partial differential equations is well known (see, e.g., [44-51]). In the present paper, the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M3">View MathML</a>

(2)

for the fractional differential equation in an arbitrary Banach space E with the linear (unbounded) operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a> is considered. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M6">View MathML</a> are the unknown and the given functions, respectively, defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M7">View MathML</a> with values in E. The derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M8">View MathML</a> is understood as the limit in the norm of E of the corresponding ratio of differences. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a> is a given closed linear operator in E with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M10">View MathML</a>, independent of t and dense in E. Finally, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M11">View MathML</a>.

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M12">View MathML</a> is the standard Riemann-Liouville derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M13">View MathML</a>. This fractional differential equation corresponds to the Basset problem [9]. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Recently, fractional Basset equations with independent in t operator coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M14">View MathML</a> have been studied extensively (see, e.g., [52-56] and the references given therein).

In the present paper, the well-posedness of problem (2) with dependent in t operator coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a> in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for 2mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of initial value problem (2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M16">View MathML</a>

(3)

is presented. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M17">View MathML</a>.

The paper is organized as follows. The well-posedness of problem (2) in spaces of smooth functions is established in Section 2. In Section 3 the coercive stability estimates for the solution of problems for 2mth order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions are obtained. The well-posedness of (3) in difference analogues of spaces of smooth functions is established and the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2mth order multidimensional fractional parabolic equation are obtained in Section 4.

2 The well-posedness of problem (2)

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a> is called a solution of problem (2) if the following conditions are satisfied:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a> is continuously differentiable on the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20">View MathML</a>. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.

(ii) The element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M22">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23">View MathML</a> and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M24">View MathML</a> is continuous on the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a> satisfies the equation and the initial condition (2).

A solution of problem (2) defined in this manner will from now on be referred to as a solution of problem (2) in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M27">View MathML</a> of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M28">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M20">View MathML</a> with values in E equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M30">View MathML</a>

(4)

In this paper, positive constants, which can differ in time, are indicated with an M. On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M31">View MathML</a> is used to focus on the fact that the constant depends only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M32">View MathML</a> .

The well-posedness in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33">View MathML</a> of boundary value problem (2) means that the coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M34">View MathML</a>

(5)

is true for its solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M35">View MathML</a>.

Suppose that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23">View MathML</a> the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M37">View MathML</a> generates an analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M38">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M39">View MathML</a>) with an exponentially decreasing norm, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M40">View MathML</a>, i.e., the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M41">View MathML</a>

(6)

hold for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M43">View MathML</a>. From this inequality it follows the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M44">View MathML</a> exists and is bounded, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33">View MathML</a>.

Suppose that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M47">View MathML</a> is Hölder continuous in t in the uniform operator topology for each fixed s, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M48">View MathML</a>

(7)

An operator-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49">View MathML</a>, defined and strongly continuous jointly in t, s for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50">View MathML</a>, is called a fundamental solution of (2) if

(1) the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49">View MathML</a> is strongly continuous in t and s for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M52">View MathML</a>,

(2)

the following identity holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M53">View MathML</a>

(3) the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49">View MathML</a> maps the region D into itself. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M55">View MathML</a> is bounded and strongly continuous in t and s for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50">View MathML</a>,

(4) on the region D the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49">View MathML</a> is strongly differentiable relative to t and s, while

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M58">View MathML</a>

(8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M59">View MathML</a>

(9)

Now, let us obtain the representation for the solution of problem (2). The initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M60">View MathML</a>

(10)

has a unique solution [54] and the following formula holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M61">View MathML</a>

(11)

Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M11">View MathML</a> and the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M63">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M64">View MathML</a>

(12)

Now, we will give a series of interesting lemmas and estimates concerning the fundamental solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M49">View MathML</a> of (2) which will be useful in the sequel.

Lemma 2.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M67">View MathML</a>, the following identities hold:

(13)

(14)

(15)

(16)

Lemma 2.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M73">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M74">View MathML</a>, the following estimates hold:

(17)

(18)

(19)

(20)

(21)

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>be a strongly positive operator in a Banach spaceEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M81">View MathML</a>. Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33">View MathML</a>of initial value problem (2), the following stability inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M84">View MathML</a>

(22)

Proof Using formula (12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M85">View MathML</a>

(23)

Applying formula (23) and the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M86">View MathML</a>

(24)

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M87">View MathML</a>

(25)

Let us first obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M88">View MathML</a>

(26)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M50">View MathML</a>. We have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M90">View MathML</a>

(27)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M91">View MathML</a>

(28)

Applying estimate (21), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M92">View MathML</a>

(29)

Now, we will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M93">View MathML</a>. We have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M94">View MathML</a>

(30)

Applying estimate (17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M95">View MathML</a>

(31)

Estimate (26) follows from estimates (29) and (31).

Now, let us first estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M96">View MathML</a>. Applying the triangle inequality and estimate (26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M97">View MathML</a>

(32)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M23">View MathML</a>. Applying the above inequality and the integral inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M99">View MathML</a>

(33)

Using the triangle inequality and equation (2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M100">View MathML</a>

(34)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M101">View MathML</a>

(35)

Estimate (22) follows from estimates (33) and (35). Theorem 2.1 is proved. □

With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>, we introduce the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>, consisting of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M105">View MathML</a> for which the following norms are finite:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M106">View MathML</a>

(36)

From (6) and (7) it follows that

Theorem 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M107">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109">View MathML</a>.

Problem (2) is not well posed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M33">View MathML</a> for arbitrary E. It turns out that a Banach space E can be restricted to a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111">View MathML</a> in such a manner that the restricted problem (2) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111">View MathML</a> will be well posed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M113">View MathML</a>. The role of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M114">View MathML</a>will be played here by the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M115">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>).

Theorem 2.3Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M117">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>). Suppose that assumptions (6) and (7) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M119">View MathML</a>. Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M121">View MathML</a>of problem (2), the coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M122">View MathML</a>

(37)

holds.

Proof

By Theorem 2.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M123">View MathML</a>

(38)

for the solution of initial value problem (2). The proof of the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M124">View MathML</a>

(39)

for the solution of initial value problem (2) is based on formula (12), estimate (38) and the following estimates [54]:

(40)

(41)

Using equation (2) and the triangle inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M127">View MathML</a>

(42)

Estimate (37) follows from estimates (39) and (42). Theorem 2.3 is proved. □

Let us give, without proof, the following result.

Theorem 2.4Suppose that assumption (6) holds. Suppose that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M47">View MathML</a>is Hölder continuous intin the uniform operator topology for each fixeds, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M129">View MathML</a>

(43)

whereMandεare positive constants independent oft, sandτfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M130">View MathML</a>. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M131">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>). Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M5">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M121">View MathML</a>of problem (2), the coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M135">View MathML</a>

(44)

holds.

3 Applications

Now, we consider the applications of Theorems 2.1, 2.3 and 2.4.

First, the Cauchy problem on the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M136">View MathML</a> for the 2m-order multidimensional fractional parabolic equation is considered:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M137">View MathML</a>

(45)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M139">View MathML</a> are given as sufficiently smooth functions. Here, σ is a sufficiently large positive constant.

Let us consider a differential operator with constant coefficients of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M140">View MathML</a>

(46)

acting on functions defined on the entire space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M142">View MathML</a> is a vector with nonnegative integer components, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M143">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M144">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M145">View MathML</a>) is an infinitely differentiable function that decays at infinity together with all its derivatives, then by means of the Fourier transformation, one establishes the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M146">View MathML</a>

(47)

Here the Fourier transform operator is defined by the following rule:

(48)

(49)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M149">View MathML</a> is called the symbol of the operator B and is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M150">View MathML</a>

(50)

We will assume that the symbol

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M151">View MathML</a>

(51)

of the differential operator of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M152">View MathML</a>

(52)

acting on functions defined on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141">View MathML</a>, satisfies the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M154">View MathML</a>

(53)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M155">View MathML</a>. Problem (45) has a unique smooth solution. This allows us to reduce problem (45) to the abstract Cauchy problem (2) in a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M156">View MathML</a> of all continuous bounded functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141">View MathML</a> satisfying the Hölder condition with the indicator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M158">View MathML</a> with a strongly positive operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M159">View MathML</a> defined by (52) (see [57,58]).

Theorem 3.1For the solution of boundary problem (45), the following estimates are satisfied:

(54)

(55)

The proof of Theorem 3.1 is based on the abstract Theorems 2.1, 2.3, 2.4 and the coercivity inequality for an elliptic operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M163">View MathML</a> and on the following theorem on the structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M164">View MathML</a>.

Theorem 3.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M165">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M166">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109">View MathML</a>[59].

Second, we consider the mixed boundary value problem for the fractional parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M168">View MathML</a>

(56)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M139">View MathML</a> are given sufficiently smooth functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M171">View MathML</a>. Here, σ is a sufficiently large positive constant.

We introduce the Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M172">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M173">View MathML</a>) of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M174">View MathML</a> satisfying the Hölder condition for which the following norms are finite:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M175">View MathML</a>

(57)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M176">View MathML</a> is the space of all continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M174">View MathML</a> defined on [0,1] with the usual norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M178">View MathML</a>

(58)

It is known that the differential expression [60]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M179">View MathML</a>

(59)

defines a positive operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162">View MathML</a> acting in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M172">View MathML</a> with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M182">View MathML</a> and satisfying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M184">View MathML</a>. Therefore, we can replace the mixed problem (56) by the abstract boundary value problem (2). Using the results of Theorems 2.1, 2.3, 2.4, we can obtain the following theorem.

Theorem 3.3For the solution of mixed problem (56), the following estimates are valid:

(60)

(61)

The proof of Theorem 3.3 is based on abstract Theorems 2.1, 2.3, 2.4 and on the following theorem on the structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M187">View MathML</a>.

Theorem 3.4<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M188">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109">View MathML</a>[60].

4 The well-posedness of problem (3)

Let us first obtain the representation for the solution of problem (3). It is clear that the first order of accuracy difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M191">View MathML</a>

(62)

has a solution and the following formula holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M192">View MathML</a>

(63)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M193">View MathML</a>

(64)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M194">View MathML</a>. Denote that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M195">View MathML</a>

(65)

Applying the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M196">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M197">View MathML</a>

(66)

So, formula (66) gives the representation for the solution of problem (3).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M198">View MathML</a> be the linear space of mesh functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M199">View MathML</a> with values in the Banach space E. Next on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M198">View MathML</a> we introduce the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M201">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M202">View MathML</a>

(67)

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>be a strongly positive operator in a Banach spaceE. Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M204">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M205">View MathML</a>of initial value problem (3), the stability inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M206">View MathML</a>

(68)

holds.

Proof Using formula (66), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M207">View MathML</a>

(69)

Applying formulas (69) and (65), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M208">View MathML</a>

(70)

Let us first obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M209">View MathML</a>

(71)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M210">View MathML</a>. We have that

(72)

Using estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M212">View MathML</a>

(73)

and the following elementary inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M213">View MathML</a>

(74)

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M214">View MathML</a>

(75)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M215">View MathML</a>

(76)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M216">View MathML</a>

(77)

Now, we will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M93">View MathML</a>. We have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M218">View MathML</a>

(78)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M219">View MathML</a>

(79)

Applying estimates (73) and (74), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M220">View MathML</a>

(80)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M221">View MathML</a>

(81)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M222">View MathML</a>

(82)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M223">View MathML</a>

(83)

Estimate (71) follows from estimates (75) and (80).

Now, let us first estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M224">View MathML</a>. Applying the triangle inequality and estimate (71), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M225">View MathML</a>

(84)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M226">View MathML</a>

(85)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M227">View MathML</a>

(86)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M228">View MathML</a>

(87)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M229">View MathML</a>. Applying the above inequality and the difference analogue of the integral inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M230">View MathML</a>

(88)

Using the triangle inequality and equation (3), we get

(89)

Estimate (68) follows from estimates (88) and (89). Theorem 4.1 is proved. □

With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>, we introduce the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>, consisting of all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M105">View MathML</a> for which the following norms are finite:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M236">View MathML</a>

(90)

From (73) it follows that

Theorem 4.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M237">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M109">View MathML</a>.

Problem (3) is not well posed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M205">View MathML</a> for arbitrary E. It turns out that a Banach space E can be restricted to a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111">View MathML</a> in such a manner that the restricted problem (3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111">View MathML</a> will be well posed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M113">View MathML</a>. The role of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M111">View MathML</a> will be played here by the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M245">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M104">View MathML</a>).

Theorem 4.3Suppose that assumptions (6) and (7) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M247">View MathML</a>. Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M248">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M249">View MathML</a>of initial value problem (3), the coercive stability inequality

(91)

holds.

Proof

By Theorem 4.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M251">View MathML</a>

(92)

for the solution of initial value problem (3). The proof of the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M252">View MathML</a>

(93)

for the solution of initial value problem (3) is based on estimate (92) and the following estimates [51]:

(94)

(95)

Using the triangle inequality and equation (3), we get

(96)

Estimate (91) follows from estimates (93) and (96). Theorem 4.3 is proved. □

Let us give, without proof, the following result.

Theorem 4.4Suppose that assumptions (6) and (43) hold. Then for the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M204">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M257">View MathML</a>of initial value problem (3), the coercive stability inequality

(97)

holds.

Note that by passing to the limit for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M259">View MathML</a>, one can recover Theorems 2.1-2.3 and 2.4.

5 Applications

Now, we consider the applications of Theorems 4.1, 4.3 and 4.4.

First, initial value problem (45) is considered. The discretization of problem (45) is carried out in two steps. In the first step, the grid space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M261">View MathML</a>) is defined as the set of all points of the Euclidean space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M141">View MathML</a> whose coordinates are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M263">View MathML</a>

(98)

The difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M264">View MathML</a> is assigned to the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M265">View MathML</a>, defined by (52). The operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M266">View MathML</a>

(99)

acts on functions defined on the entire space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M268">View MathML</a> is a vector with nonnegative integer coordinates,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M269">View MathML</a>

(100)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M270">View MathML</a> is the unit vector of the axis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M271">View MathML</a>.

An infinitely differentiable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M272">View MathML</a> of the continuous argument <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M273">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.boundaryvalueproblems.com/content/2013/1/31/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274">View MathML</a> is a λth order (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M275">View MathML</a>) approximation of the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162">View MathML</a> if the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M277">View MathML</a>

(101)

holds for any smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M278">View MathML</a>. The coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M279">View MathML</a> are chosen in such a way that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274">View MathML</a> approximates in a specified way the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162">View MathML</a>. It will be assumed that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274">View MathML</a> approximates the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M162">View MathML</a> with any prescribed order [57,58].

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M284">View MathML</a> is obtained by replacing the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M285">View MathML</a> in the right-hand side of equality (99) with the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M286">View MathML</a>, respectively, and is called the symbol of the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M287">View MathML</a>.

It will be assumed that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M288">View MathML</a> and fixed x, the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M289">View MathML</a> of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M290">View MathML</a> satisfies the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M291">View MathML</a>

(102)

Suppose that the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M292">View MathML</a> of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M290">View MathML</a> is bounded and satisfies the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M294">View MathML</a>

(103)

With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274">View MathML</a>, we arrive at the nonlocal boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M296">View MathML</a>

(104)

for an infinite system of ordinary differential equations.

In the second step, problem (104) is replaced by the difference scheme

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M297">View MathML</a>

(105)

Based on the number of corollaries of the abstract theorems given in the above, to formulate the result, one needs to introduce the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M299">View MathML</a> of all bounded grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M300">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M260">View MathML</a>, equipped with the norms

(106)

(107)

Theorem 5.1Suppose that assumptions (102) and (103) for the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304">View MathML</a>hold. Then, the solutions of difference scheme (105) satisfy the following stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M305">View MathML</a>

(108)

The proof of Theorem 5.1 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306">View MathML</a> defined by (114) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M307">View MathML</a> and on the following two theorems on the coercivity inequality for the solution of the elliptic difference equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M308">View MathML</a> and on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M309">View MathML</a>.

Theorem 5.2Suppose that assumptions (102) and (103) for the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304">View MathML</a>hold. Then for the solutions of the elliptic difference equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M311">View MathML</a>

(109)

the estimates[54]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M312">View MathML</a>

(110)

are valid.

Theorem 5.3Suppose that assumptions (102) and (103) for the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M304">View MathML</a>hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M314">View MathML</a>, the norms in the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M309">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M316">View MathML</a>are equivalent uniformly inh[51].

Second, we consider mixed boundary value problem (56). The discretization of problem (56) is carried out in two steps. In the first step, let us define the grid space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M317">View MathML</a>

(111)

We introduce the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M318">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M173">View MathML</a>) of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M320">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M321">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M322">View MathML</a>

(112)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M323">View MathML</a> is the space of the grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M324">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M321">View MathML</a>, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M326">View MathML</a>

(113)

To the differential operator A generated by problem (56), we assign the difference operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M328">View MathML</a>

(114)

acting in the space of grid functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M329">View MathML</a> satisfying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M330">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M331">View MathML</a>. With the help of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M306">View MathML</a>, we arrive at the initial boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M333">View MathML</a>

(115)

for an infinite system of ordinary fractional differential equations. In the second step, we replace problem (115) by difference scheme (3)

(116)

Theorem 5.4Letτandhbe sufficiently small numbers. Then, the solutions of difference scheme (116) satisfy the following stability estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M335">View MathML</a>

(117)

The proof of Theorem 5.4 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M274">View MathML</a> defined by (114) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M307">View MathML</a> and on the following theorem on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M338">View MathML</a>.

Theorem 5.5For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M189">View MathML</a>, the norms in the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M340">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M341">View MathML</a>are equivalent uniformly inhand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M342">View MathML</a>[60].

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank Prof. P. E. Sobolevskii for his helpful suggestions to the improvement of this paper.

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