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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Linear and nonlinear convolution elliptic equations

Veli B Shakhmurov12 and Ismail Ekincioglu3*

Author Affiliations

1 Department of Mechanical Engineering, Okan University, Tuzla, Istanbul, Turkey

2 Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

3 Department of Mathematics, Dumlupınar University, Kütahya, Turkey

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Boundary Value Problems 2013, 2013:211  doi:10.1186/1687-2770-2013-211

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/211


Received:16 May 2013
Accepted:31 July 2013
Published:19 September 2013

© 2013 Shakhmurov and Ekincioglu; 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 this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1">View MathML</a> spaces. In application, maximal regularity properties of anisotropic elliptic convolution equations are studied.

MSC: 34G10, 45J05, 45K05.

Keywords:
positive operators; Banach-valued spaces; operator-valued multipliers; boundary value problems; convolution equations; nonlinear integro-differential equations

1 Introduction

In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptic-type, have been studied extensively, e.g., in [1-13] and the references therein (for comprehensive references, see [13]). Moreover, in [14,15], on embedding theorems and maximal regular differential operator equations in Banach-valued function spaces have been studied. Also, in [16,17], on theorems on the multiplicators of Fourier integrals obtained, which were used in studying isotropic as well as anisotropic spaces of differentiable functions of many variables. In addition, multiplicators of Fourier integrals for the spaces of Banach valued functions were studied. On the basis of these results, embedding theorems are proved.

Moreover, convolution-differential equations (CDEs) have been treated, e.g., in [1,18-22] and [23]. Convolution operators in vector valued spaces are studied, e.g., in [24-26] and [27]. However, the convolution-differential operator equations (CDOEs) are a relatively less investigated subject (see [13]). The main aim of the present paper is to establish the separability properties of the linear CDOE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M2">View MathML</a>

(1.1)

and the existence and uniqueness of the following nonlinear CDOE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M3">View MathML</a>

in E-valued <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1">View MathML</a> spaces, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5">View MathML</a> is a possible unbounded operator in a Banach space E, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M6">View MathML</a> are complex-valued functions, and λ is a complex parameter. We prove that the problem (1.1) has a unique solution u, and the following coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M7">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>. The methods are based on operator-valued multiplier theorems, theory of elliptic operators, vector-valued convolution integrals, operator theory and etc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in [1].

2 Notations and background

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M11">View MathML</a> denote the space of all strongly measurable E-valued functions that are defined on the measurable subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M12">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M13">View MathML</a>

Let C be the set of complex numbers, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M14">View MathML</a>

A linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M16">View MathML</a> is said to be uniformly positive in a Banach space E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M17">View MathML</a> is dense in E, does not depend on x, and there is a positive constant M so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M18">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M21">View MathML</a>, where I is an identity operator in E, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M22">View MathML</a> is the space of all bounded linear operators in E, equipped with the usual uniform operator topology. Sometimes, instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M23">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M24">View MathML</a> and denote it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M25">View MathML</a>. It is known (see [28], §1.14.1) that there exist fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M26">View MathML</a>of the positive operator A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M27">View MathML</a> denote the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M28">View MathML</a> with the graphical norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M29">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M30">View MathML</a> denote Schwartz class, i.e., the space of E-valued rapidly decreasing smooth functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31">View MathML</a>, equipped with its usual topology generated by semi-norms. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M32">View MathML</a> denoted by just S. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M33">View MathML</a> denote the space of all continuous linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M34">View MathML</a>, equipped with the bounded convergence topology. Recall <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M30">View MathML</a> is norm dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M38">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M39">View MathML</a> are integers. An E-valued generalized function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M40">View MathML</a> is called a generalized derivative in the sense of Schwartz distributions of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M41">View MathML</a> if the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M42">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M43">View MathML</a>.

Let F denote the Fourier transform. Through this section, the Fourier transformation of a function f will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M44">View MathML</a>. It is known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M45">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M46">View MathML</a>.

Let Ω be a domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M49">View MathML</a> will denote the spaces of E-valued bounded uniformly strongly continuous and m-times continuously differentiable functions on Ω, respectively. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M50">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M49">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M52">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M54">View MathML</a> are two Banach spaces. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M55">View MathML</a> is called a multiplier from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M56">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M57">View MathML</a> if the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M59">View MathML</a> is well defined and extends to a bounded linear operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M60">View MathML</a>

Let Q denotes a set of some parameters. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M61">View MathML</a> be a collection of multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M62">View MathML</a>. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M63">View MathML</a> is a collection of uniformly bounded multipliers (UBM) if there exists a positive constant M independent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M65">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M59">View MathML</a>.

A Banach space E is called an UMD-space [29,30] if the Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M68">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70">View MathML</a>[29]. The UMD spaces include, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M72">View MathML</a> spaces and Lorentz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M74">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M75">View MathML</a> is called R-bounded (see [5,6,12]) if there is a positive constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M76">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M80">View MathML</a> is a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M81">View MathML</a>-valued random variables on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M82">View MathML</a>. The smallest C, for which the above estimate holds, is called an R-bound of the collection W and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M83">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M84">View MathML</a>, dependent on parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64">View MathML</a>, is called uniformly R-bounded with respect to h if there is a positive constant C, independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M64">View MathML</a>, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M89">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M90">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M91">View MathML</a>.

Definition 2.1 A Banach space E is said to be a space, satisfying the multiplier condition, if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M92">View MathML</a> the R-boundedness of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M93">View MathML</a>

implies that Ψ is a Fourier multiplier, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M94">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M95">View MathML</a>.

The uniform R-boundedness of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M96">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M97">View MathML</a>

implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M98">View MathML</a> is a uniformly bounded collection of Fourier multipliers (UBM) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36">View MathML</a>.

Remark 2.2 Note that if E is UMD space, then by virtue of [5,7,12,25], it satisfies the multiplier condition. The UMD spaces satisfy the uniform multiplier condition (see Proposition 2.4).

Definition 2.3 A positive operator A is said to be a uniformly R-positive in a Banach space E if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M21">View MathML</a> such that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M101">View MathML</a>

is uniformly R-bounded.

Note that every norm bounded set in Hilbert spaces is R-bounded. Therefore, all sectorial operators in Hilbert spaces are R-positive.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M105">View MathML</a> be standard unit vectors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M107">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a> be a closed linear operator in E with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110">View MathML</a> independent of x. The Fourier transformation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M111">View MathML</a> is a linear operator with the same domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110">View MathML</a> defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M113">View MathML</a>

(For details see [[2], p.7].) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5">View MathML</a> be a closed linear operator in E with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110">View MathML</a> independent of x. Then, it is differentiable if there is the limit

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M116">View MathML</a>

in the sense of E-norm.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a> be closed linear operator in E with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M110">View MathML</a> independent of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M120">View MathML</a>. We can define the convolution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M121">View MathML</a> in the distribution sense by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M122">View MathML</a>

(see [2]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123">View MathML</a> and E be two Banach spaces, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123">View MathML</a> is continuously and densely embedded into E. Let l be a integer number. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M125">View MathML</a> denote the space of all functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M127">View MathML</a> and the generalized derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M128">View MathML</a> with the following norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M129">View MathML</a>

It is clearly seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M130">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M131">View MathML</a> satisfying the equation (1.1) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M31">View MathML</a>, is called a solution of equation (1.1).

The elliptic CDOE (1.1) is said to be separable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M133">View MathML</a> if for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M134">View MathML</a> the equation (1.1) has a unique solution u, and the following coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M135">View MathML</a>

where the constant C do not depend on f.

In a similar way as Theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M136">View MathML</a> in [31], Theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M136">View MathML</a> and by reasoning as Theorem 3.7 in [7], we obtain the following.

Proposition 2.4LetEbeUMDspace, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M138">View MathML</a>and suppose there is a positive constantKsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M139">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M98">View MathML</a>is UBM in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9">View MathML</a>.

Proof Really, some steps of proof trivially work for the parameter dependent case (see [7]). Other steps can be easily shown by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M143">View MathML</a>

instead of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M144">View MathML</a>

and by using uniformly R-boundedness of set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M145">View MathML</a>. However, parameter depended analog of Proposition 3.4 in [7] is not straightforward. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M147">View MathML</a> be Fourier multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M133">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M149">View MathML</a> converge to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M146">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M151">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M152">View MathML</a> be uniformly bounded with respect to h and N. Then by reasoning as Proposition 3.4 in [7], we obtain that the operator function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M153">View MathML</a> is uniformly bounded with respect to h. Hence, by using steps above, in a similar way as Theorem 3.7 in [7], we obtain the assertion.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M54">View MathML</a> be two Banach spaces. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M158">View MathML</a> will denote operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a>. □

In a similar way as Proposition 2.11 in [12], we have

Proposition 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M37">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M75">View MathML</a>isR-bounded, then the collection<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M164">View MathML</a>is alsoR-bounded.

From [11], we obtain the following.

Theorem 2.6Let the following conditions be satisfied

1. Eis a Banach space satisfying the uniform multiplier condition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M166">View MathML</a>are certain parameters;

2. lis a positive integer, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M167">View MathML</a>aren-tuples of nonnegative integer numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M169">View MathML</a>;

3. Ais anR-positive operator inEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M170">View MathML</a>.

Then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M171">View MathML</a>is continuous, and there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M172">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M173">View MathML</a>

Theorem 2.7Let the following conditions be satisfied

1. Eis a Banach space satisfying the uniform multiplier condition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M166">View MathML</a>are certain parameters;

2. lis a positive integer, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M167">View MathML</a>aren-tuples of nonnegative integer numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M169">View MathML</a>;

3. Ais anR-positive operator inEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M170">View MathML</a>.

Then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M180">View MathML</a>is continuous, and there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M181">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M182">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M183">View MathML</a>.

3 Elliptic CDOE

Condition 3.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M184">View MathML</a> and the following hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M185">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M187">View MathML</a>.

In the following, we denote the operator functions by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189">View MathML</a>.

Lemma 3.2Assume Condition 3.1 holds, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190">View MathML</a>is a uniformlyφ-positive operator inEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M191">View MathML</a>. Then, the following operator functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M192">View MathML</a>

are uniformly bounded, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M193">View MathML</a>.

Proof By virtue of Lemma 2.3 in [4] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M196">View MathML</a> there is a positive constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M197">View MathML</a>

(3.1)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M198">View MathML</a>, in view of (3.1) and resolvent properties of positive operators, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M199">View MathML</a> is invertible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M200">View MathML</a>

Next, let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M201">View MathML</a>. It is clearly seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M202">View MathML</a>

(3.2)

Since A is uniformly φ-positive and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M198">View MathML</a>, then setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M204">View MathML</a> in the following well-known inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M205">View MathML</a>

(3.3)

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M206">View MathML</a>

Taking into account the Condition 3.1 and (3.1)-(3.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M207">View MathML</a>

 □

Lemma 3.3Assume Condition 3.1 holds, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M208">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190">View MathML</a>be a uniformlyφ-positive operator in a Banach spaceEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M211">View MathML</a>and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M212">View MathML</a>

(3.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M213">View MathML</a>

(3.5)

Then, operator functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M214">View MathML</a>are uniformly bounded.

Proof Let us first prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M215">View MathML</a> is uniformly bounded. Really,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M216">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M217">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M218">View MathML</a>

By using (3.1) and (3.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M219">View MathML</a>

Due to positivity of A, by using (3.1) and (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M220">View MathML</a>

Since, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M190">View MathML</a> is uniformly φ-positive, by using (3.1), (3.3) and (3.4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M222">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M223">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M224">View MathML</a>

In a similar way, the uniform boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M225">View MathML</a> is proved. Next, we shall prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M226">View MathML</a> is uniformly bounded. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M227">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M228">View MathML</a>

Let us first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M229">View MathML</a> is uniformly bounded. It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M230">View MathML</a>

Due to positivity of A, by virtue of (3.1) and (3.3)-(3.5), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M231">View MathML</a>. In a similar way, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M232">View MathML</a>. Hence, operator functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189">View MathML</a> are uniformly bounded. From the representations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235">View MathML</a>, it easy to see that operator functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M236">View MathML</a> contain similar terms as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M237">View MathML</a>, namely, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M238">View MathML</a> will be represented as combinations of principal terms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M239">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M240">View MathML</a>. Therefore, by using similar arguments as above and in view of (3.6), one can easily check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M241">View MathML</a>

 □

Lemma 3.4Let all conditions of the Lemma 3.2 hold. Suppose thatEis a Banach space satisfying the uniform multiplier condition, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M242">View MathML</a>is a uniformlyRpositive operator in E. Then, the following sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M243">View MathML</a>

are uniformlyR-bounded for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M244">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245">View MathML</a>.

Proof Due to R-positivity of A we obtain that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M246">View MathML</a>

is R bounded. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M247">View MathML</a>

the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M248">View MathML</a> is R -bounded. Moreover, in view of Condition 3.1 and (3.1), there is a positive constant M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M249">View MathML</a>

Then, by virtue of Kahane’s contraction principle, Lemma 3.5 in [5], we obtain that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M250">View MathML</a> is uniformly R-bounded. Then by Lemma 3.2, we obtain the uniform R-boundedness of sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M251">View MathML</a>, i.e,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M252">View MathML</a>

(3.7)

Moreover, due to boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M253">View MathML</a>, in view of Condition 3.1 and by virtue of (3.1) and (3.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M254">View MathML</a>

(3.8)

In view of representation (3.6) and estimate (3.8), we need to show uniform R-boundedness of the following sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M255">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M240">View MathML</a>. By virtue of Kahane’s contraction principle, additional and product properties of R-bounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in [5], and in view of (3.7), it is sufficient to prove uniform R-boundedness of the following set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M257">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M258">View MathML</a>

thanks to R-boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M259">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M260">View MathML</a>

(3.9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M79">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M80">View MathML</a> is a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M81">View MathML</a>-valued random variables on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M267">View MathML</a>. Thus, in view of Kahane’s contraction principle, additional and product properties of R-bounded operators and (3.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M268">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M269">View MathML</a>

(3.11)

The estimate (3.10) implies R-boundedness of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M270">View MathML</a>. Moreover, from Lemma 3.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M271">View MathML</a>

i.e., we obtain the assertion. □

The following result is the corollary of Lemma 3.4 and Proposition 2.4.

Result 3.5Suppose that all conditions of Lemma 3.3 are satisfied, EisUMDspace, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M272">View MathML</a>is a uniformlyR-positive operator inE. Then the sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M189">View MathML</a>are uniformlyR-bounded.

Now, we are ready to present our main results. We find sufficient conditions that guarantee separability of problem (1.1).

Condition 3.6 Suppose that the following are satisfied

1. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M186">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M278">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245">View MathML</a>;

3. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M284">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M285">View MathML</a>

Theorem 3.7Suppose that Condition 3.6 holds, andEis a Banach space satisfying the uniform multiplier condition. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M286">View MathML</a>be a uniformlyR-positive inEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M191">View MathML</a>. Then, problem (1.1) has a unique solutionu, and the following coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M288">View MathML</a>

(3.12)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M290">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>.

Proof By applying the Fourier transform to equation (1.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M292">View MathML</a>

Hence, the solution of equation (1.1) can be represented as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M293">View MathML</a>. Then there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M296">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235">View MathML</a> are operator functions defined in Lemma 3.3. Therefore, it is sufficient to show that the operator-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M235">View MathML</a> are UBM in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M36">View MathML</a>. However, these follow from Lemma 3.4. Thus, from (3.13), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M300">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8">View MathML</a>. Hence, we get assertion.

Let O be an operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M302">View MathML</a> that is generated by the problem (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M303">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M304">View MathML</a>

 □

Result 3.8Theorem 2.6 implies that the operatorOis separable inX, i.e., for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M305">View MathML</a>, all terms of equation (1.1) also are fromX, and for solutionuof equation (1.1), there are positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a>so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M308">View MathML</a>

Condition 3.9 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M309">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M310">View MathML</a>. Moreover, there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a> so that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M315">View MathML</a>

Remark 3.10 Condition 3.9 is checked for the regular elliptic operators with smooth coefficients on sufficiently smooth domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M316">View MathML</a> considered in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M318">View MathML</a> (see Theorem 5.1).

Theorem 3.11Assume that all conditions of Theorem 3.7 and Condition 3.9 are satisfied. LetEbe a Banach space satisfying the uniform multiplier condition. Then, problem (1.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M319">View MathML</a>, and the following coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M320">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M322">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M323">View MathML</a>.

Proof By applying the Fourier transform to equation (1.1), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M324">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M325">View MathML</a>

So, we obtain that the solution of equation (1.1) can be represented as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M293">View MathML</a>. Moreover, by Condition 3.9, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M327">View MathML</a>

Hence, by using estimates (3.12), it is sufficient to show that the operator functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M329">View MathML</a> are UBM in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M330">View MathML</a>. Really, in view of Condition 3.9, and uniformly R-positivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M286">View MathML</a>, these are proved by reasoning as in Lemma 3.4. □

Condition 3.12 There are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M334">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M276">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M336">View MathML</a>

in cases, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M337">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M338">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M339">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M340">View MathML</a>.

Theorem 3.13Let all conditions of Theorem 3.11 and Condition 3.12 hold. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M341">View MathML</a>, there are positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M342">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M343">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M344">View MathML</a>

Proof The left part of the inequality above is derived from Theorem 3.11. So, it remains to prove the right side of the estimate. Really, from Condition 3.12 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M345">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M346">View MathML</a>

Hence, applying the Fourier transform to equation (1.1), and by reasoning as Theorem 3.11, it is sufficient to prove that the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M347">View MathML</a>

is a multiplier in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M348">View MathML</a>. In fact, by using Condition 3.12 and the proof of Lemma 3.2, we get desired result. □

Result 3.14Theorem 3.13 implies that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M349">View MathML</a>, there are positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M352">View MathML</a>

From Theorem 3.7, we have the following.

Result 3.15Assume all conditions of Theorem 3.7 hold. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>, the resolvent of operatorOexists, and the following sharp estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M354">View MathML</a>

Result 3.16Theorem 3.7 particularly implies that the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M355">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M356">View MathML</a>is positive in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M348">View MathML</a>, i.e., if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M358">View MathML</a>is uniformlyR-positive for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M359">View MathML</a>, then (see, e.g., [28], §1.14.5) the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M360">View MathML</a>is a generator of an analytic semigroup in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M361">View MathML</a>.

From Theorems 3.7, 3.11, 3.13 and Proposition 2.4, we obtain the following.

Result 3.17Let conditions of Theorems 3.7, 3.11, 3.13 hold for Banach spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M362">View MathML</a>, respectively. Then assertions of Theorems 3.7, 3.11, 3.13 are valid.

4 The quasilinear CDOE

Consider the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M363">View MathML</a>

(4.1)

in E-valued <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M1">View MathML</a> spaces, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M5">View MathML</a> is a possible unbounded operator in Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M366">View MathML</a> are complex-valued functions, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M367">View MathML</a> denote all differential operators that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M368">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M369">View MathML</a>

Remark 4.1 By using Theorem 2.7, we obtain that the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M370">View MathML</a> is continuous, and by trace theorem [32] (or [19]) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M371">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M373">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M374">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M375">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M376">View MathML</a> denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M377">View MathML</a>. Consider the linear CDOE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M378">View MathML</a>

(4.2)

From Theorem 3.7, we conclude that problem (4.2) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M379">View MathML</a>, and the coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M380">View MathML</a>

(4.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M9">View MathML</a>.

Condition 4.2 Assume that all conditions of Theorem 3.11 are satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M383">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M384">View MathML</a>. Suppose that

1. The function: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M385">View MathML</a> is a Lipschitz function from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M123">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M387">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M388">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M390">View MathML</a> is a measurable function for each u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M391">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M392">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M393">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M394">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M395">View MathML</a> is continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M397">View MathML</a>. Moreover, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M398">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M399">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M402">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M403">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M404">View MathML</a>.

Theorem 4.3Let Condition 4.2 hold. Then, there exist a radius<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M405">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M406">View MathML</a>such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M407">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M408">View MathML</a>there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M409">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M410">View MathML</a>satisfying equation (3.13).

Proof We want to to solve problem (4.1) locally by means of maximal regularity of the linear problem (4.2) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (4.2). Consider the following ball

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M411">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M412">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M413">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M414">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M415">View MathML</a>.

Define a map G on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M417">View MathML</a>

(4.4)

where u is a solution of problem (4.1). We want to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M418">View MathML</a>, and that L is a contraction operator in Y. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M419">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M420">View MathML</a>, moreover, δ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M421">View MathML</a> can be chosen such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M422">View MathML</a>. In fact, since by Theorem 2.7, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M423">View MathML</a>, and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M424">View MathML</a>

Thus, Q is measurable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M425">View MathML</a>

Now, by Remark 4.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M426">View MathML</a>, by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M427">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M428">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M429">View MathML</a>

Moreover, by Theorem 3.11 and by embedding Theorem 2.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M430">View MathML</a>

Thus, G maps the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M416">View MathML</a>. Let us show that G is a strict contraction. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M433">View MathML</a>

It is clearly seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M434">View MathML</a> is a solution of the linear problem (4.2) for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M435">View MathML</a>

Then, by using estimate (4.3) and reasoning as above, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M436">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M437">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M438">View MathML</a>, we obtain that G is a strict contraction. Then by virtue of contraction mapping principle, we obtain that problem (4.1) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M341">View MathML</a>. □

5 Boundary value problems for integro-differential equations

In this section, by applying Theorem 3.7, the BVP for the anisotropic type convolution equations is studied. The maximal regularity of this problem in mixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M440">View MathML</a> norms is derived. In this direction, we can mention, e.g., the works [2,18,21] and [33].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M441">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M442">View MathML</a> is an open connected set with a compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M443">View MathML</a>-boundary Ω. Consider the BVP for integro-differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M444">View MathML</a>

(5.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M445">View MathML</a>

(5.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M446">View MathML</a>

In general, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M447">View MathML</a>, so equation (4.4) is anisotropic. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M448">View MathML</a>, we get isotropic equation. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M449">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M450">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M451">View MathML</a> will denote the space of all p-summable scalar-valued functions with a mixed norm (see, e.g., [34]), i.e., the space of all measurable functions f defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M452">View MathML</a>, for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M453">View MathML</a>

Analogously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M454">View MathML</a> denotes the Sobolev space with a corresponding mixed norm [34]. Let Q denote the operator, generated by problem (4.4) and (5.1). In this section, we present the following result.

Theorem 5.1Let the following conditions be satisfied

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M455">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M456">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M457">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M458">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M459">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M460">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M461">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M462">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M463">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M464">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M467">View MathML</a>;

3. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M468">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M469">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M470">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M472">View MathML</a>let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M473">View MathML</a>;

4. For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M474">View MathML</a>local BVP in local coordinates corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M475">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M476">View MathML</a>

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M477">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M478">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M479">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M480">View MathML</a>;

5. The (1) part of Condition 3.6 is satisfied, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M481">View MathML</a>, and there are positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M483">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M484">View MathML</a>

Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M485">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>problems (4.4) and (5.1) have a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M487">View MathML</a>, and the following coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M488">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317">View MathML</a>. It is known [29] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490">View MathML</a> is UMD space for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M318">View MathML</a>. Consider the operator A in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M493">View MathML</a>

(5.3)

Therefore, problems (4.4) and (5.1) can be rewritten in the form of (1.1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M495">View MathML</a> are functions with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M317">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M497">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M498">View MathML</a> are operators in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M500">View MathML</a>

(5.4)

In view of conditions and by [[5], Theorem 8.2] operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M501">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M502">View MathML</a> for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M503">View MathML</a>, are uniformly R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M490">View MathML</a>. Moreover, by (3.3), the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M505">View MathML</a>

(5.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M506">View MathML</a>

(5.6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M507">View MathML</a> and for sufficiently large μ, have unique solutions that belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M508">View MathML</a>, and the coercive estimates hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M509">View MathML</a>

for solutions of problems (5.4) and (5.5). Then in view of (5) condition and by virtue of embedding theorems [34], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M510">View MathML</a>

(5.7)

Moreover by using (5) condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M511">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M512">View MathML</a>

i.e., all conditions of Theorem 3.7 hold, and we obtain the assertion. □

6 Infinite system of IDEs

Consider the following infinity system of a convolution equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M513">View MathML</a>

(6.1)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M515">View MathML</a> .

Condition 6.1 There are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M295">View MathML</a>, so that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M518">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M109">View MathML</a> and some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M520">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M521">View MathML</a>

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M522">View MathML</a>, and there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M523">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M483">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M525">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M526">View MathML</a>

Let Q be a differential operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M527">View MathML</a>, generated by problem (5.7) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M528">View MathML</a>. Applying Theorem 3.7, we have the following.

Theorem 6.2Suppose that (1) part of Condition 3.6 and Condition 6.1 are satisfied. Then

1. For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M529">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M467">View MathML</a>the equation (6.1) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M532">View MathML</a>that belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M533">View MathML</a>, and the coercive uniform estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M534">View MathML</a>

2. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M10">View MathML</a>, there exists a resolvent<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M536">View MathML</a>of operatorQand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M537">View MathML</a>

Proof Really, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M538">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M539">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M540">View MathML</a> . Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M541">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M497">View MathML</a> is uniformly R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/211/mathml/M543">View MathML</a>, and all conditions of Theorem 3.7 are hold. Therefore, by virtue of Theorem 3.7 and Result 4.1, we obtain the assertions. □

Remark 6.3 There are a lot of positive operators in concrete Banach spaces. Therefore, putting concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of operator A in (1.1) and (4.1), we can obtain the maximal regularity of different class of convolution equations, Cauchy problems for parabolic CDEs or it’s systems, by virtue of Theorem 3.7 and Theorem 3.11, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper.

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