This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Singular degenerate problems occurring in biosorption process

Aida Sahmurova1 and Veli B Shakhmurov2*

Author Affiliations

1 Department of Environmental Sciences, Okan University, Akfirat, Tuzla, Istanbul, 34959, Turkey

2 Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul, 34959, Turkey

For all author emails, please log on.

Boundary Value Problems 2013, 2013:30  doi:10.1186/1687-2770-2013-30


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/30


Received:22 November 2012
Accepted:5 February 2013
Published:15 February 2013

© 2013 Sahmurova and Shakhmurov; licensee Springer

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

Abstract

The boundary value problems for singular degenerate arbitrary order differential-operator equations with variable coefficients are considered. The uniform coercivity properties of ordinary and partial differential equations with small parameters are derived in abstract <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a> spaces. It is shown that corresponding differential operators are positive and also are generators of analytic semigroups. In application, well-posedeness of the Cauchy problem for an abstract parabolic equation and systems of parabolic equations are studied in mixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M2">View MathML</a> spaces. These problems occur in fluid mechanics and environmental engineering.

MSC: 34G10, 35J25, 35J70.

Keywords:
differential-operator equations; degenerate equations; semigroups of operators; Banach-valued function spaces; coercive problems; operator-valued Fourier multipliers; interpolation of Banach spaces

0 Introduction

Boundary value problems (BVPs) for differential-operator equations (DOEs) in H-valued (Hilbert space valued) function spaces have been studied extensively by many researchers (see, e.g., [1-14] and the references therein). A comprehensive introduction to DOEs and historical references may be found in [6] and [14]. The maximal regularity properties for DOEs have been studied, e.g., in [3,10-19].

In this work, singular degenerate BVPs for arbitrary order DOEs with parameters are considered. This problem has numerous applications. The parameter-dependent BVPs occur in different situations of fluid mechanics and environmental engineering etc.

In Section 2, the BVP for the following singular degenerate ordinary DOE with a small parameter:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M3">View MathML</a>

is considered, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M4">View MathML</a>

t is a small parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M5">View MathML</a> is a complex-valued function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M7">View MathML</a> is a principal, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M9">View MathML</a> are subordinate linear operators in a Banach space E. Several conditions for the uniform coercivity and the resolvent estimates for this problem are given in abstract <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a>-spaces. We prove that the problem has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14">View MathML</a> with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a> and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M16">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M17">View MathML</a>

In Section 3, the partial DOE with small parameters

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M18">View MathML</a>

is considered in a mixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M19">View MathML</a> space, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M20">View MathML</a> are complex-valued functions, A and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M21">View MathML</a> are linear operators in E, λ is a complex and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22">View MathML</a> are positive parameters, G is an n-dimensional rectangular domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M23">View MathML</a>. Here we prove that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a> with sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, this problem has a unique solution u that belongs to the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M27">View MathML</a> with a mixed p norm and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M28">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M29">View MathML</a>

In Section 4, the uniform well-posedeness of the mixed problem for the following singular degenerate abstract parabolic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M30">View MathML</a>

is obtained. Particularly, the above problem occurs in atmospheric dispersion of pollutants and evolution models for phytoremediation of metals from soils. In application, particularly, by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M34">View MathML</a>, we consider the mixed problem for the system of the following parabolic equations with parameters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M35">View MathML</a>

which arises in phytoremediation process, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M36">View MathML</a> are real-valued functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M37">View MathML</a> are data. The maximal regularity properties of this problem are studied. Note that the maximal regularity properties for undegenerate DOEs were investigated, e.g., in [1-10,14-16,19,20]. Regular degenerate DOEs in Banach spaces were treated in [11-13,15,17-19,21]. It should be noted that contrary to these results, here high-order singular degenerated BVPs with small parameters are considered. Moreover, principal coefficients depend on space variables. The proofs are based on abstract harmonic analysis, operator theory, interpolation of Banach spaces, theory of semigroups of linear operators, microlocal analysis, embedding and trace theorems in vector-valued Sobolev-Lions spaces.

1 Notations and background

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M39">View MathML</a> be a positive measurable function on a domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M40">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M41">View MathML</a> denotes the space of strongly measurable E-valued functions that are defined on Ω with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M42">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M43">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M44">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M45">View MathML</a>.

The Banach space E is called a UMD-space if the Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M46">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M48">View MathML</a> (see, e.g., [22]). UMD spaces include, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M50">View MathML</a> spaces and Lorentz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M52">View MathML</a>.

Let C denote the set of complex numbers and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M53">View MathML</a>

A linear operator A is said to be φ-positive in a Banach space E with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M54">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M55">View MathML</a> is dense on E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M56">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14">View MathML</a>, where I is the identity operator in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M59">View MathML</a> is the space of bounded linear operators in E. Sometimes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M60">View MathML</a> will be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M61">View MathML</a> and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M62">View MathML</a>. It is known [23], Section 1.15.1] that a positive operator A has well-defined fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M63">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M64">View MathML</a> denote the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M65">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M66">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M68">View MathML</a> be two Banach spaces continuously embedded in a locally convex space. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M71">View MathML</a>, we denote the interpolation spaces obtained from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M72">View MathML</a> by the K-method [23], Section 1.3.2].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M73">View MathML</a> denote the space of E-valued uniformly bounded continuous functions on the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M40">View MathML</a>.

Let N denote the set of natural numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M75">View MathML</a> be a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M76">View MathML</a>-valued random variables on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M77">View MathML</a> (see [22]). A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M78">View MathML</a> is called uniform R-bounded with respect to h (see, e.g., [16]) if there is a constant C independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M79">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M83">View MathML</a>

The smallest C for which the above estimate holds is called an R-bound of the collection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M84">View MathML</a> and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M85">View MathML</a>.

A φ-positive operator A is said to be R-positive in E if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M14">View MathML</a>, is R-bounded.

Note that for Hilbert spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M89">View MathML</a>, all norm-bounded sets are R-bounded (see, e.g., [16]). Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M91">View MathML</a>, or A has the bounded imaginary powers with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M93">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M94">View MathML</a>, then those operators are R-positive (e.g., see [16], Section 4.3]).

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M96">View MathML</a> is said to be φ-positive in E uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M97">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M98">View MathML</a> is independent of t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M99">View MathML</a> is dense in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M100">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M102">View MathML</a>, where M does not depend on t.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103">View MathML</a> and E be two Banach spaces and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103">View MathML</a> be continuously and densely embedded into E. Let m be a positive integer. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M105">View MathML</a> denotes an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M103">View MathML</a>-valued function space defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M107">View MathML</a>

Let t be a positive parameter. We define a parameterized norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M108">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M109">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M112">View MathML</a> denote the space of all p-summable E-valued functions with a mixed norm (see, e.g., [24], Section 8] for scalar case), i.e., the space of all measurable E-valued functions f defined on G, for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M113">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M114">View MathML</a> be positive integers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22">View MathML</a> be positive parameters and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M117">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M118">View MathML</a>

Consider the following weighted spaces of functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M119">View MathML</a>

with the mixed norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M120">View MathML</a>

and with the parameterized norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M121">View MathML</a>

respectively.

Consider the BVP for DOE

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M122">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M125">View MathML</a> are complex numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M127">View MathML</a>; A is a possible unbounded operator in E.

In a similar way as in [17], Theorem 5.1], we obtain the following.

Theorem A1Let the following conditions be satisfied:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M129">View MathML</a>are complex numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M130">View MathML</a>, tis a small positive parameter and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M131">View MathML</a>;

(2) Eis a UMD space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M133">View MathML</a>;

(3) Ais anR-positive operator inE.

Then problem (1) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a>with sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M137">View MathML</a>. Moreover, the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M138">View MathML</a>

By reasoning as in [17], Theorem 2.3], we obtain the following.

Theorem A2Let the following conditions be satisfied:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M142">View MathML</a>;

(2) Eis a UMD space andAis anR-positive operator inE;

(3) there exists a bounded linear extension operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M143">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M144">View MathML</a>.

Then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M145">View MathML</a>is continuous and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M146">View MathML</a>, the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M147">View MathML</a>

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M148">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M149">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M150">View MathML</a>

Theorem A3Let the following conditions be satisfied:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M153">View MathML</a>;

(2) Eis a UMD space andAis anR-positive operator inE;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M155">View MathML</a>aren-tuples of a nonnegative integer such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M156">View MathML</a>

(4) there exists a bounded linear extension operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M157">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M158">View MathML</a>.

Then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M159">View MathML</a>is continuous. Moreover, there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M160">View MathML</a>such that for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M161">View MathML</a>, the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M162">View MathML</a>

2 Singular degenerate DOEs with parameter

Consider the BVP for the following differential-operator equation with parameter:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M163">View MathML</a>

(2)

on the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164">View MathML</a>, where t is a positive parameter and λ is a complex parameter; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M125">View MathML</a> are complex numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M5">View MathML</a> is a complex-valued function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M77">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M172">View MathML</a> are linear operators in a Banach space E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M173">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M174">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M175">View MathML</a> satisfying equation (2) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164">View MathML</a> is said to be the solution of equation (2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M164">View MathML</a>.

Remark 1

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M178">View MathML</a>

(3)

Under the substitution (3), spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M180">View MathML</a> are mapped isomorphically onto weighted spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M181">View MathML</a>

respectively, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M182">View MathML</a>

Moreover, under the substitution (3), problem (2) is transformed into the following non-degenerate problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M183">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M184">View MathML</a>

(5)

in the weighted space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M185">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M189">View MathML</a> are again denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M192">View MathML</a>, γ after the substitution (3), respectively.

Let us consider boundary value problem (4)-(5).

Theorem 1Let the following conditions be satisfied:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M190">View MathML</a>is a positive uniformly bounded continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195">View MathML</a>;

(2) Eis a UMD space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M133">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M198">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M191">View MathML</a>isR-positive inEuniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M200">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M202">View MathML</a>;

(4) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203">View MathML</a>, there is a positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M205">View MathML</a>

Then problem (4)-(5) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M206">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M207">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a>with sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>. Moreover, the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M210">View MathML</a>

(6)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M211">View MathML</a> be bounded intervals in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M213">View MathML</a> correspond to a partition of unit that functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M214">View MathML</a> are smooth on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M217">View MathML</a>. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M218">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M219">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M220">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M221">View MathML</a>, from (4) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M222">View MathML</a>

(7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M223">View MathML</a>

(8)

Since a is uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195">View MathML</a> for all small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M225">View MathML</a>, there is a large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M226">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M227">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M228">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M229">View MathML</a>

Cover <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M230">View MathML</a> by finitely many intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M231">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M232">View MathML</a>

Define coefficients of local operators, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M233">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M234">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M235">View MathML</a> . Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M237">View MathML</a> , we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M238">View MathML</a>

Freezing coefficients in (7) obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M239">View MathML</a>

(9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M240">View MathML</a>

(10)

Since functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M241">View MathML</a> have compact supports in (9), if we extend <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M241">View MathML</a> on the outsides of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M243">View MathML</a>, we obtain BVPs with constant coefficients

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M244">View MathML</a>

(11)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M245">View MathML</a> denote E-valued weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a>-norms with respect to domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M247">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M214">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M249">View MathML</a>. Then, by virtue of Theorem A1, we obtain that problem (11) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M253">View MathML</a>

(12)

Theorem A2 implies that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203">View MathML</a>, there is a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M256">View MathML</a>

(13)

Consequently, by using Theorem A2, from (12)-(13) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M257">View MathML</a>

(14)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M258">View MathML</a>, from (14) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M259">View MathML</a>

(15)

Then, by using the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M260">View MathML</a> and by virtue of (15) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M261">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M262">View MathML</a>

(16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M263">View MathML</a> be a solution of problem (4)-(5). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M265">View MathML</a>

(17)

By Theorem A2, by virtue of (16) and (17) for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M267">View MathML</a>

(18)

Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M269">View MathML</a> generated by problem (4)-(5), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M270">View MathML</a>

Estimate (18) implies that problem (4)-(5) has only a unique solution and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M271">View MathML</a> has an invertible operator in its rank space. We need to show that this rank space coincides with the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M272">View MathML</a>. We consider the smooth functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M273">View MathML</a> with respect to the partition of the unit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M274">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M195">View MathML</a> that equals one on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M276">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M278">View MathML</a>. Let us construct for all j the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250">View MathML</a> that is defined on the regions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M280">View MathML</a> and satisfies problem (4)-(5). Problem (4)-(5) can be expressed in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M281">View MathML</a>

(19)

Consider operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M282">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M283">View MathML</a> generated by BVPs (19). By virtue of Theorem A1 for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M284">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M287">View MathML</a>

(20)

Extending <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M250">View MathML</a> zero on the outside of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M276">View MathML</a> in equalities (20) and passing substitutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M290">View MathML</a>, we obtain operator equations with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M291">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M292">View MathML</a>

(21)

By virtue of Theorem A2, by estimate (20), in view of the smoothness of the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M293">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M296">View MathML</a>, where ε is sufficiently small. Consequently, equations (21) have unique solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M297">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M298">View MathML</a>

Whence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M299">View MathML</a> are bounded linear operators from X to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M300">View MathML</a>. Thus, we obtain that the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M301">View MathML</a>

are the solutions of equations (21). Consider the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M302">View MathML</a> in X such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M303">View MathML</a>

It is clear from the constructions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M304">View MathML</a> and estimate (20) that operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M305">View MathML</a> are bounded linear from X to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M306">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M307">View MathML</a>

(22)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M302">View MathML</a> is a bounded linear operator from X to X. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309">View MathML</a> denote the operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M310">View MathML</a> generated by BVP (4)-(5). Then the act of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M311">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M312">View MathML</a> gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M313">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M314">View MathML</a> is a linear combination of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M315">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M316">View MathML</a> . By virtue of Theorem A2, estimate (22) and in view of the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M317">View MathML</a>, we obtain that operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M318">View MathML</a> are bounded linear from X to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M283">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M320">View MathML</a>. Therefore, there exists a bounded linear invertible operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M321">View MathML</a>. Whence, we obtain that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M322">View MathML</a>, BVP (4)-(5) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M323">View MathML</a>

(23)

Then, by using the above representation and in view of Theorem A1, we obtain the assertion of Theorem 1. □

Result 1 Theorem 1 implies that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309">View MathML</a> has a resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M325">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294">View MathML</a> and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M327">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M328">View MathML</a> denote the operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M329">View MathML</a> generated by BVP (2). By virtue of Theorem 1 and Remark 1, we obtain the following result.

Result 2 Let all conditions of Theorem 1 be satisfied. Then

(a) problem (2) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M330">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M12">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>. Moreover, the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M333">View MathML</a>

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M328">View MathML</a> has a resolvent operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M335">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M336">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M337">View MathML</a>

Theorem 2Let all conditions of Theorem 1 hold. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309">View MathML</a>is uniformlyR-positive in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M179">View MathML</a>, also<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M309">View MathML</a>is a generator of an analytic semigroup.

Proof By virtue of Theorem 1, we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M341">View MathML</a>, BVP (4)-(5) has a unique solution expressed in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M342">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M343">View MathML</a> are local operators generated by problems (7)-(8) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M345">View MathML</a> are uniformly bounded operators in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M346">View MathML</a>. In a similar way as in [1,11,17], we obtain that operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M347">View MathML</a> are R-positive. Then, by using the above representation and by virtue of Kahane’s contraction principle, the product and additional properties of the collection of R-bounded operators (see, e.g., [16], Lemma 3.5, Proposition 3.4), we obtain the assertions. □

3 Singular degenerate anisotropic equation with parameters

Consider the following degenerate BVP with parameters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M348">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M350">View MathML</a> are linear operators in a Banach space E,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M351">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M352">View MathML</a> are complex-valued functions on G, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M353">View MathML</a> are complex numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22">View MathML</a> are positive and λ is a complex parameter.

Note that BVP (24) is degenerated with different speeds on different directions in general.

The main result of this section is the following.

Theorem 3Assume the following conditions hold:

(1) Eis a UMD space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M171">View MathML</a>isR-positive inEuniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M356">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M357">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M358">View MathML</a>;

(2) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M203">View MathML</a>, there is a positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M204">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M361">View MathML</a>

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M363">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M365">View MathML</a>;

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M352">View MathML</a>are continuous positive functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M367">View MathML</a>.

Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294">View MathML</a>and sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, problem (24) has a unique solutionuthat belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M371">View MathML</a>and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M372">View MathML</a>

(25)

Proof

Consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M373">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M374">View MathML</a> are boundary conditions of type (24) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M375">View MathML</a>. By virtue of Result 2, problem (26) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M376">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M377">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>. Moreover, the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M380">View MathML</a>

Let us now consider in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M381">View MathML</a> the BVP on the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M382">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M383">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M384">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M385">View MathML</a>, then problem (27) can be expressed in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M386">View MathML</a>

where B is the differential operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M387">View MathML</a> generated by problem (26), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M388">View MathML</a>

By virtue of [22], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M389">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M390">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M94">View MathML</a>. Moreover, by virtue of Theorem 2, the operator B is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M392">View MathML</a>. Hence, by Result 2, we get that problem (27) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M393">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M394">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M294">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, and coercive uniform estimate (25) holds. By continuing this process for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M397">View MathML</a>, we obtain that the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M398">View MathML</a>

(28)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M13">View MathML</a> and sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M15">View MathML</a>, has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M402">View MathML</a> and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M403">View MathML</a>

(29)

Moreover, by virtue of embedding Theorem A3, we have the Ehrling-Nirenberg-Gagliardo type estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M404">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405">View MathML</a> denote the operator generated by problem (28) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M406">View MathML</a>

By using estimate (29), we obtain that there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M407">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M408">View MathML</a>

Then, by using perturbation elements, we obtain the assertion. □

From Theorem 2 and Theorem 3, we obtain the following result.

Result 3 Let all conditions of Theorem 3 hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M409">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M410">View MathML</a>. Then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405">View MathML</a> is uniformly R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M412">View MathML</a>, it also is a generator of an analytic semigroup.

4 Singular degenerate parabolic DOE

Consider the following mixed problem for a parabolic DOE with parameter:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M413">View MathML</a>

(30)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M415">View MathML</a>, G, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M417">View MathML</a> are defined as in Section 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M419">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M420">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M421">View MathML</a> will denote the space of all E-valued <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M422">View MathML</a>-summable functions with a mixed norm. Analogously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M423">View MathML</a>

denotes the Sobolev space with a corresponding mixed norm (see [24] for a scalar case).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405">View MathML</a> denote a differential operator generated by (28) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M425">View MathML</a>.

Theorem 4Let all conditions of Theorem 3 hold for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M426">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M427">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M428">View MathML</a>and sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418">View MathML</a>, problem (30) has a unique solution belonging to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M430">View MathML</a>and the following coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M431">View MathML</a>

Proof Problem (30) can be expressed as the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M432">View MathML</a>

(31)

Result 3 implies that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405">View MathML</a> is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M434">View MathML</a>. By [23], Section 1.14], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M405">View MathML</a> is a generator of an analytic semigroup in F. Then, by virtue of [20], Theorem 4.2], we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M436">View MathML</a> problem (31) has a unique solution belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M437">View MathML</a> and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M438">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M439">View MathML</a>, by Theorem 3 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M440">View MathML</a>

The above estimate proves the hypothesis to be true. □

5 Cauchy problem for infinite systems of degenerate parabolic equations with small parameters

Consider the infinity systems of BVP for the degenerate anisotropic parabolic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M441">View MathML</a>

(32)

where N is finite or infinite natural number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M443">View MathML</a>, G, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M417">View MathML</a>, d are defined as in Sections 3 and 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M447">View MathML</a> are real functions and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M448">View MathML</a>

From Theorem 4 we obtain the following.

Theorem 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M450">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M451">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M452">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M453">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M454">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M455">View MathML</a>and sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M418">View MathML</a>, problem (32) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M457">View MathML</a>that belongs to the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M458">View MathML</a>and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M459">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M460">View MathML</a> and A be infinite matrices such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M461">View MathML</a>

It is clear that the operator A is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M462">View MathML</a>. Problem (32) can be rewritten as problem (30). Then, from Theorem 4, we obtain the assertion. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The results to proving the uniform coercivity properties of ordinary and partial differential equations with small parameters in abstract <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a> spaces, the showing that corresponding differential operators are positive and also are generators of analytic semigroups and well-posedeness of Cauchy problem for abstract parabolic equation and systems of parabolic equations are studied in mixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M2">View MathML</a> spaces due to VS.

The applications of these abstract problems to concrete mathematics and engineering problem belongs to AS.

Both authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

References

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

  2. Ashyralyev, A: On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim.. 24(1 & 2), 1–15 (2003)

  3. Dore, C, Yakubov, S: Semigroup estimates and non coercive boundary value problems. Semigroup Forum. 60, 93–121 (2000). Publisher Full Text OpenURL

  4. Favini, A, Shakhmurov, V, Yakubov, Y: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup Forum. 79(1), 22–54 (2009). Publisher Full Text OpenURL

  5. Goldstain, JA: Semigroups of Linear Operators and Applications, Oxford University Press, Oxford (1985)

  6. Krein, SG: Linear Differential Equations in Banach Space, Am. Math. Soc., Providence (1971)

  7. Lunardi, A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (2003)

  8. Lions, J-L, Magenes, E: Nonhomogenous Boundary Value Problems, Mir, Moscow (1971)

  9. Sobolevskii, PE: Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR. 57(1), 27–40 (1964)

  10. Prüss, J: Maximal regularity for evolution equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a>-spaces. Monopoli. (2002)

  11. Shakhmurov, VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Theory Methods Appl.. 73, 2383–2397 (2010). Publisher Full Text OpenURL

  12. Shakhmurov, VB: Imbedding theorems and their applications to degenerate equations. Differ. Equ.. 24(4), 475–482 (1988)

  13. Shakhmurov, VB: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl.. 292(2), 605–620 (2004). Publisher Full Text OpenURL

  14. Yakubov, S, Yakubov, Y: Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000)

  15. Amann, H: Linear and Quasi-Linear Equations, Birkhäuser, Basel (1995)

  16. Denk, R, Hieber, M, Prüss, J: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788) (2003)

  17. Shakhmurov, VB: Degenerate differential operators with parameters. Abstr. Appl. Anal.. 2006, 1–27 (2007)

  18. Shakhmurov, VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl.. 327(2), 1182–1201 (2006)

  19. Shakhmurov, VB: Nonlinear abstract boundary value problems in vector-valued function spaces and applications. Nonlinear Anal., Theory Methods Appl.. 67(3), 745–762 (2006)

  20. Weis, L: Operator-valued Fourier multiplier theorems and maximal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/30/mathml/M1">View MathML</a> regularity. Math. Ann.. 319, 735–758 (2001). Publisher Full Text OpenURL

  21. Agarwal, R, O’Regan, D, Shakhmurov, VB: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl.. 338, 970–983 (2008). Publisher Full Text OpenURL

  22. Burkholder, DL: A geometrical condition that implies the existence certain singular integral of Banach space-valued functions. Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, pp. 270–286. Wadsworth, Belmont Chicago, 1981. (1983)

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

  24. Besov, OV, Ilin, VP, Nikolskii, SM: Integral Representations of Functions and Embedding Theorems, Nauka, Moscow (1975)