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This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Allaberen Ashyralyev12, Sema Akturk1 and Yasar Sozen13*

Author Affiliations

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

2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan

3 Present address: Fen Fakultesi, Matematik Bolumu, Hacettepe Universitesi, 06800, Beytepe, Ankara, Turkey

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Boundary Value Problems 2014, 2014:3  doi:10.1186/1687-2770-2014-3

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/3


Received:30 October 2013
Accepted:3 December 2013
Published:2 January 2014

© 2014 Ashyralyev et al.; 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

We consider the two-dimensional differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M1">View MathML</a> defined on functions on the half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M2">View MathML</a> with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6">View MathML</a>, are continuously differentiable and satisfy the uniform ellipticity condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M8">View MathML</a>. The structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M9">View MathML</a> generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.

MSC: 35J25, 47E05, 34B27.

Keywords:
positive operator; fractional spaces; Green’s function; Hölder spaces

1 Introduction

It is well known that (see, for example, [1-3] and the references therein) various classical and non-classical boundary value problems for partial differential equations can be considered as an abstract boundary value problem for an ordinary differential equation in a Banach space with a densely defined unbounded operator. The importance of the positivity property of the differential operators in a Banach space in the study of various properties for partial differential equations is well known (see, for example, [4-7] and the references therein). Several authors have investigated the positivity of a wider class of differential and difference operators in Banach spaces (see [8-18] and the references therein).

Let us give the definition of positive operators and introduce the fractional spaces and preliminary facts that will be needed in the sequel.

The operator A is said to be positive in E if its spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M10">View MathML</a> lies inside of the sector S of the angle ϕ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M11">View MathML</a>, symmetric with respect to the real axis, and the following estimate (see, for example, [6,19])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M12">View MathML</a>

holds on the edges <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M14">View MathML</a> of S, and outside of the sector S. The infimum of all such angles ϕ is called the spectral angle of the positive operator A and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M15">View MathML</a>. We say that A is a strongly positive operator in E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M16">View MathML</a>.

Throughout the article, M indicates positive constants which may differ from time to time, and we are not interested to precise. If the constant depends only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M17">View MathML</a> , then we will write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M18">View MathML</a>.

With the help of the positive operator A, we introduce the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M19">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M20">View MathML</a>), consisting of all elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M21">View MathML</a> for which the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M22">View MathML</a>

is finite.

Theorem 1[20]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M24">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M25">View MathML</a>be any two nonnegative integrable functions such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M27">View MathML</a>. Then the following Hilbert’s inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M28">View MathML</a>

Danelich in [12] considered the positivity of a difference analog <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M29">View MathML</a> of the 2mth-order multi-dimensional elliptic operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M30">View MathML</a> with dependent coefficients on semi-spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M31">View MathML</a>.

The structure of fractional spaces generated by positive multi-dimensional differential and difference operators on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M32">View MathML</a> in Banach spaces has been well investigated (see [21-23] and the references therein).

In papers [19,24-27] the structure of fractional spaces generated by positive one-dimensional differential and difference operators in Banach spaces was studied. Note that the structure of fractional spaces generated by positive multi-dimensional differential and difference operators with local and nonlocal conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M33">View MathML</a> in Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M34">View MathML</a> has not been well studied.

In the present paper, we study the structure of fractional spaces generated by the two-dimensional differential operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M35">View MathML</a>

(1)

defined over the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M36">View MathML</a> with the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4">View MathML</a>. Here, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6">View MathML</a>, are continuously differentiable and satisfy the uniform ellipticity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M41">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M42">View MathML</a>.

Following the paper [12], passing limit when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M43">View MathML</a> in the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M45">View MathML</a>, we get that there exists the inverse operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M46">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47">View MathML</a> and the following formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M48">View MathML</a>

(2)

holds, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M49">View MathML</a> is the Green function of differential operator (1). Moreover, the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M50">View MathML</a>

(3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M51">View MathML</a>

(4)

hold. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M52">View MathML</a>.

Next, to formulate our result, we need to introduce the Hölder space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M53">View MathML</a> of all continuous bounded functions φ defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M54">View MathML</a> satisfying a Hölder condition with the indicator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M55">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M56">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M57">View MathML</a> denotes the Banach space of all continuous bounded functions φ defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M54">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M59">View MathML</a>

Clearly, from estimates (3) and (4) it follows that A is a positive operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M60">View MathML</a>. Namely, we have the following.

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47">View MathML</a>. Then the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M62">View MathML</a>

is valid.

Here, the structure of fractional spaces generated by the operator A is investigated. The positivity of A in Hölder spaces is studied. The organization of the present paper is as follows. In Section 2, the positivity of A in Hölder spaces is established. In Section 3, the main theorem on the structure of fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M63">View MathML</a> generated by A is investigated. In Section 4, applications on theorems on well-posedness in a Hölder space of parabolic and elliptic problems are presented. Finally, the conclusion is given.

2 Positivity of A in Hölder spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M64">View MathML</a>

Theorem 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M65">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47">View MathML</a>, we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M67">View MathML</a>

Proof Applying formula (2), the triangle inequality, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68">View MathML</a>-norm, estimate (3), and Hilbert’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M69">View MathML</a>

Then from that it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M70">View MathML</a>

(5)

Without loss of generality, we can put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M71">View MathML</a>. Using formula (2) and the triangle inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M72">View MathML</a>

(6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M73">View MathML</a>. Now, we will estimate the right-hand side of inequality (6). Let us consider two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a> separately. First, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>. Using the triangle inequality, estimate (4), the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68">View MathML</a>-norm, Hilbert’s inequality, and the Lagrange theorem, it follows that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M84">View MathML</a>

(7)

Second, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>. Using formula (2), the triangle inequality, estimate (3), the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M68">View MathML</a>-norm, and estimate (6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M87">View MathML</a>

(8)

Estimates (7) and (8) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M88">View MathML</a>

(9)

Combining estimates (5) and (9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M89">View MathML</a>

This finishes the proof of Theorem 3. □

Note that from the commutativity of A and its resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M90">View MathML</a>, and Theorem 3, we have the following theorem.

Theorem 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M47">View MathML</a>. Then the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M92">View MathML</a>

3 The structure of fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M93">View MathML</a>

Suppose β, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M94">View MathML</a>. Consider the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M95">View MathML</a> and the Hölder space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M96">View MathML</a>. In this section, we prove the following structure theorem.

Theorem 5The norms of the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M98">View MathML</a>are equivalent.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M99">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101">View MathML</a> be fixed. From formula (2) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M102">View MathML</a>

(10)

Using equation (10), the triangle inequality, the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M103">View MathML</a>

(11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M104">View MathML</a>

(12)

estimates (10), (11), (12), and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105">View MathML</a>-norm, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M106">View MathML</a>

(13)

Thus, it follows from estimate (13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M107">View MathML</a>

(14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100">View MathML</a> be fixed. Using equation (10), we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M110">View MathML</a>

(15)

Now, we will estimate the right-hand side of equation (15). We consider two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>, respectively. Let us first assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>. Furthermore, this situation will be considered in two cases: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M115">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M114">View MathML</a>. From equation (15), the triangle inequality, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105">View MathML</a>-norm, the assumptions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M119">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M120">View MathML</a>

(16)

We will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M122">View MathML</a>, separately.

First, let us estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M123">View MathML</a>. Clearly, by the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M124">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M125">View MathML</a>

(17)

From estimates (3), (11), (12), and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M126">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M127">View MathML</a>

(18)

Estimates (3), (11), (12), and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M124">View MathML</a> yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M129">View MathML</a>

(19)

Combining estimates (17)-(19), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M130">View MathML</a>

(20)

Now, let us consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M115">View MathML</a>. Then, using equation (10), we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M132">View MathML</a>

(21)

From equation (21), the triangle inequality, the Lagrange theorem, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M105">View MathML</a>-norm, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, it follows that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M135">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M137">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M141">View MathML</a>

We will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M143">View MathML</a>, separately. Let us start with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M144">View MathML</a>. Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M145">View MathML</a>

(22)

Using estimates (3), (4), (11), (12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M146">View MathML</a>

(23)

Note that by the triangle inequality and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M150">View MathML</a>

(24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M151">View MathML</a>

(25)

By inequalities (11), (12), estimates (4), (24), (25), and Hilbert’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M152">View MathML</a>

(26)

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M153">View MathML</a>

(27)

Combining estimates (22), (23), (26), (27), we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M155">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M156">View MathML</a>

(28)

It follows from estimates (20) and (28) that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M159">View MathML</a>

(29)

Next, let us assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>. By equation (10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M161">View MathML</a>

(30)

Equation (30), estimates (3), (11), Hilbert’s inequality, the triangle inequality, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M162">View MathML</a>-norm, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a> yield that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M101">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M165">View MathML</a>

(31)

From estimates (29) and (31) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M166">View MathML</a>

(32)

Combining estimates (14) and (32), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M167">View MathML</a>

Now, we will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M168">View MathML</a>. By Theorem 4, A is a positive operator in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169">View MathML</a>. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M171">View MathML</a>

(33)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M172">View MathML</a>. It follows from formula (2) and equation (33) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M173">View MathML</a>

(34)

Using the triangle inequality, equation (34), estimate (3), and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169">View MathML</a>-norm, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M175">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M176">View MathML</a>

(35)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M100">View MathML</a> be fixed. From equation (34) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M178">View MathML</a>

(36)

Now, we will estimate the right-hand side of equation (36). We consider two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>. Let us first assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>. By equation (36), the triangle inequality, the Lagrange theorem, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M169">View MathML</a>-norm, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, we have, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M78">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M80">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M81">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M83">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M190">View MathML</a>

We will estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M122">View MathML</a>, separately. Using the triangle inequality, estimates (4), (11), (25), (24), (12), the Lagrange theorem, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M194">View MathML</a>

From the triangle inequality, estimates (4), (11), (12), (24), (25), the Lagrange theorem, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M196">View MathML</a>

Using the triangle inequality, estimates (3), (24), (25), (12), (11), the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, and the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M198">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M199">View MathML</a>

Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M74">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M201">View MathML</a>

(37)

Next, let us assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>. Using formula (36), estimate (3), the triangle inequality, Hilbert’s inequality, and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M204">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M75">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M206">View MathML</a>

(38)

Combining estimates (37) and (38), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M207">View MathML</a>

(39)

Estimates (35) and (39) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M208">View MathML</a>

This is the end of the proof of Theorem 5. □

4 Applications

In this section, we consider some applications of Theorem 5. First, we consider the boundary value problem for the elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M209">View MathML</a>

(40)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M213">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M214">View MathML</a> are given smooth functions and they satisfy every compatibility condition and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M215">View MathML</a>

(41)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M42">View MathML</a>, which guarantees that problem (40) has a smooth solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M217">View MathML</a>.

Theorem 6For the solution of boundary value problem (40), we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M218">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M219">View MathML</a>is independent ofφ, ψ, andf.

Proof We introduce the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M220">View MathML</a> of all continuous abstract functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M221">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M222">View MathML</a> with values in E, equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M223">View MathML</a>

Note that problem (40) can be written in the form of the abstract boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M224">View MathML</a>

(42)

in a Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M225">View MathML</a> with a positive operator A defined by (1). Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M226">View MathML</a> is the given abstract function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M222">View MathML</a> with values in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M229">View MathML</a> are elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M230">View MathML</a>. Therefore, the proof of Theorem 6 is based on Theorem 5 on the structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M63">View MathML</a>, Theorem 4 on the positivity of the operator A, on the following theorems on coercive stability of elliptic problems, nonlocal boundary value for the abstract elliptic equation and on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M232">View MathML</a>. This is the end of the proof of Theorem 6. □

Theorem 7Under assumption (41) for the solution of elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M233">View MathML</a>

the following coercive inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M234">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M235">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M236">View MathML</a>) does not depend ong.

The proof of Theorem 7 uses the techniques introduced in [[5], Chapter 5] and it is based on estimates (3) and (4).

Theorem 8 ([[5], Theorem 5.2.48])

The spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M237">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M238">View MathML</a>coincide for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M239">View MathML</a>, and their norms are equivalent.

Theorem 9 ([[28], Theorem 3.1])

LetAbe a positive operator in a Banach spaceEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M240">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M241">View MathML</a>). Then, for the solution of nonlocal boundary value problem (42), the coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M242">View MathML</a>

holds, whereMdoes not depend onα, φ, ψ, andf.

Second, we consider the nonlocal boundary value problem for the elliptic equation under assumption (41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M243">View MathML</a>

(43)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M211">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M214">View MathML</a> are given smooth functions and they satisfy every compatibility condition and (41), which guarantees that problem (43) has a smooth solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M217">View MathML</a>.

Theorem 10For the solution of initial boundary value problem (43), we have the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M248">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M219">View MathML</a>is independent off.

The proof of Theorem 10 is based on Theorem 5 on the structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M250">View MathML</a>, Theorem 4 on the positivity of the operator A, Theorem 7 on coercive stability of the elliptic problem, Theorem 8 on the structure of the fractional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M232">View MathML</a>, and the following theorem on coercive stability of the nonlocal boundary value for the abstract elliptic equation.

Theorem 11 ([[28], Theorem 3.1])

LetAbe a positive operator in a Banach spaceEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M252">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M241">View MathML</a>). Then, for the solution of nonlocal boundary value problem (40)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M254">View MathML</a>

in a Banach space E with a positive operatorA, the coercive inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M255">View MathML</a>

holds, whereMdoes not depend onαandf.

5 Conclusion

In the present article, the structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M95">View MathML</a> generated by the two-dimensional elliptic differential operator A is investigated. The positivity of this operator A in a Hölder space is established. Of course, the Banach fixed point theorem and the method of the present paper enable us to establish the existence and uniqueness results which hold under some sufficient conditions on the nonlinear term for the solution of the mixed problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M257">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Some of the results of the present article were announced in the conference proceeding [11] as an extended abstract without proofs. The authors would like to thank Prof. Pavel Sobolevskii (Universidade Federal do Ceará, Brasil). The second author would also like to thank The Scientific and Technological Research Council of Turkey (TUBİTAK) for the financial support.

References

  1. Fattorini, HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland, Amsterdam (1985)

  2. Grisvard, P: Elliptic Problems in Nonsmooth Domains, Pitman, London (1984)

  3. Krein, SG: Linear Differential Equations in a Banach Space, Am. Math. Soc., Providence (1968)

  4. Ashyralyev, A, Sobolevskii, PE: Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basel (1994)

  5. Ashyralyev, A, Sobolevskii, PE: New Difference Schemes for Partial Differential Equations, Birkhäuser, Basel (2004)

  6. Krasnosel’skii, MA, Zabreiko, PP, Pustyl’nik, EI, Sobolevskii, PE: Integral Operators in Spaces of Summable Functions, Noordhoff, Leiden (1976)

  7. Agarwal, R, Bohner, M, Shakhmurov, VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl.. 1, 9–42 (2005)

  8. Alibekov, KhA, Sobolevskii, PE: Stability of difference schemes for parabolic equations. Dokl. Akad. Nauk SSSR. 232(4), 737–740 (1977)

  9. Alibekov, KhA: Investigations in C and Lp of difference schemes of high order accuracy for approximate solutions of multidimensional parabolic boundary value problems. Dissertation. Voronezh State University, Voronezh (1978)

  10. Alibekov, KhA, Sobolevskii, PE: Stability and convergence of difference schemes of a high order for parabolic differential equations. Ukr. Mat. Zh.. 31(6), 627–634 (1979)

  11. Ashyralyev, A, Akturk, S, Sozen, Y: Positivity of two-dimensional elliptic differential operators in Hölder space. AIP Conf. Proc.. 1470, 77–79 (2012)

  12. Danelich, SI: Fractional powers of positive difference operators. Dissertation. Voronezh State University, Voronezh (1989)

  13. Lunardi, A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (1995)

  14. Simirnitskii, YuA, Sobolevskii, PE: Positivity of multidimensional difference operators in the C-norm. Usp. Mat. Nauk. 36(4), 202–203 (1981)

  15. Sobolevskii, PE: The coercive solvability of difference equations. Dokl. Akad. Nauk SSSR. 201(5), 1063–1066 (1980)

  16. Solomyak, MZ: Analytic semigroups generated by elliptic operator in spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M259">View MathML</a>. Dokl. Akad. Nauk SSSR. 127(1), 37–39 (1959)

  17. Solomyak, MZ: Estimation of norm of the resolvent of elliptic operator in spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M261">View MathML</a>. Usp. Mat. Nauk. 15(6), 141–148 (1960)

  18. Stewart, HB: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc.. 259, 299–310 (1980). Publisher Full Text OpenURL

  19. Ashyralyev, A: Fractional spaces generated by the positive differential and difference operators in a Banach space. In: Tas K, Tenreiro Machado JA, Baleanu D (eds.) Mathematical Methods in Engineering, Springer, Dordrecht (2007)

  20. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities, Cambridge University Press, Cambridge (1988)

  21. Ashyralyev, A, Sobolevskii, PE: The linear operator interpolation theory and the stability of the difference schemes. Dokl. Akad. Nauk SSSR. 275(6), 1289–1291 (1984)

  22. Triebel, H: Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978)

  23. Ashyralyev, A: Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations. Dissertation, Inst. of Math. of Acad. Sci. Kiev (1992)

  24. Ashyralyev, A, Yaz, N: On structure of fractional spaces generated by positive operators with the nonlocal boundary value conditions. In: Agarwal RP (ed.) Proceedings of the Conference Differential and Difference Equations and Applications, Hindawi Publishing Corporation, New York (2006)

  25. Ashyralyev, A, Tetikoğlu, FS: The structure of fractional spaces generated by the positive operator with periodic conditions. (2012)

  26. Bazarov, MA: On the structure of fractional spaces. Proceedings of the XXVII All-Union Scientific Student Conference ‘The Student and Scientific-Technological Progress’, pp. 3–7. Novosibirsk. Gos. Univ., Novosibirsk (1989) (in Russian)

  27. Ashyralyev, A, Nalbant, N, Sozen, Y: Structure of fractional spaces generated by second order difference operators. J. Franklin Inst. (2013). Publisher Full Text OpenURL

  28. Ashyralyev, A: On well-posedness of the nonlocal boundary value problems for elliptic equations. Numer. Funct. Anal. Optim.. 24, 1–15 (2003). Publisher Full Text OpenURL