Open Access

Degenerate Anisotropic Differential Operators and Applications

Boundary Value Problems20112011:268032

DOI: 10.1155/2011/268032

Received: 2 December 2010

Accepted: 18 January 2011

Published: 23 February 2011

Abstract

The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banach-valued https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq1_HTML.gif spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. In the last section, some applications of the main results are given.

1. Introduction and Notations

It is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed as differential-operator equations (DOEs). As a result, many authors investigated PDEs as a result of single DOEs. DOEs in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq2_HTML.gif -valued (Hilbert space valued) function spaces have been studied extensively in the literature (see [114] and the references therein). Maximal regularity properties for higher-order degenerate anisotropic DOEs with constant coefficients and nondegenerate equations with variable coefficients were studied in [15, 16].

The main aim of the present paper is to discuss the separability properties of BVPs for higher-order degenerate DOEs; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq4_HTML.gif are weighted functions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq6_HTML.gif are linear operators in a Banach Space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq7_HTML.gif . The above DOE is a generalized form of an elliptic equation. In fact, the special case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq9_HTML.gif reduces (1.1) to elliptic form.

Note, the principal part of the corresponding differential operator is nonself-adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness, discreetness of the spectrum, and completeness of root elements of this operator are established.

We prove that the corresponding differential operator is separable in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq10_HTML.gif ; that is, it has a bounded inverse from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq11_HTML.gif to the anisotropic weighted space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq12_HTML.gif . This fact allows us to derive some significant spectral properties of the differential operator. For the exposition of differential equations with bounded or unbounded operator coefficients in Banach-valued function spaces, we refer the reader to [8, 1525].

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq13_HTML.gif be a positive measurable weighted function on the region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq14_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq15_HTML.gif denote the space of all strongly measurable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq16_HTML.gif -valued functions that are defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq17_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ2_HTML.gif
(1.2)

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq18_HTML.gif , the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq19_HTML.gif will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq20_HTML.gif .

The weight https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq21_HTML.gif we will consider satisfies an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq22_HTML.gif condition; that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq24_HTML.gif if there is a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq25_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ3_HTML.gif
(1.3)

for all cubes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq26_HTML.gif .

The Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq27_HTML.gif is called a UMD space if the Hilbert operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq28_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq30_HTML.gif (see, e.g., [26]). UMD spaces include, for example, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq32_HTML.gif spaces, and Lorentz spaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq35_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq36_HTML.gif be the set of complex numbers and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ4_HTML.gif
(1.4)
A linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq37_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq38_HTML.gif -positive in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq39_HTML.gif with bound https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq40_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq41_HTML.gif is dense on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq42_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ5_HTML.gif
(1.5)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq44_HTML.gif is an identity operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq45_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq46_HTML.gif is the space of bounded linear operators in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq47_HTML.gif . Sometimes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq48_HTML.gif will be written as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq49_HTML.gif and denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq50_HTML.gif . It is known [27, Section 1.15.1] that there exists fractional powers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq51_HTML.gif of the sectorial operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq52_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq53_HTML.gif denote the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq54_HTML.gif with graphical norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ6_HTML.gif
(1.6)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq56_HTML.gif be two Banach spaces. Now, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq58_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq59_HTML.gif will denote interpolation spaces obtained from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq60_HTML.gif by the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq61_HTML.gif method [27, Section 1.3.1].

A set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq62_HTML.gif is called https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq63_HTML.gif -bounded (see [3, 25, 26]) if there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq64_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq67_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq68_HTML.gif is a sequence of independent symmetric https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq69_HTML.gif -valued random variables on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq70_HTML.gif .

The smallest https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq71_HTML.gif for which the above estimate holds is called an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq72_HTML.gif -bound of the collection https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq73_HTML.gif and is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq74_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq75_HTML.gif denote the Schwartz class, that is, the space of all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq76_HTML.gif -valued rapidly decreasing smooth functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq77_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq78_HTML.gif be the Fourier transformation. A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq79_HTML.gif is called a Fourier multiplier in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq80_HTML.gif if the map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq81_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq82_HTML.gif is well defined and extends to a bounded linear operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq83_HTML.gif . The set of all multipliers in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq84_HTML.gif will denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq85_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ8_HTML.gif
(1.8)

Definition 1.1.

A Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq86_HTML.gif is said to be a space satisfying a multiplier condition if, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq87_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq88_HTML.gif -boundedness of the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq89_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq90_HTML.gif is a Fourier multiplier in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq91_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq92_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq93_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq94_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq95_HTML.gif be a multiplier function dependent on the parameter https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq96_HTML.gif . The uniform https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq97_HTML.gif -boundedness of the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq98_HTML.gif ; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ9_HTML.gif
(1.9)

implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq99_HTML.gif is a uniform collection of Fourier multipliers.

Definition 1.2.

The https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq100_HTML.gif -positive operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq101_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq102_HTML.gif -positive in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq103_HTML.gif if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq104_HTML.gif such that the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq105_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq106_HTML.gif -bounded.

A linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq107_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq108_HTML.gif -positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq109_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq110_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq111_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq113_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq115_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq117_HTML.gif .

The https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq118_HTML.gif -positive operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq120_HTML.gif is said to be uniformly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq121_HTML.gif -positive in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq122_HTML.gif if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq123_HTML.gif such that the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq124_HTML.gif is uniformly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq125_HTML.gif -bounded; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ10_HTML.gif
(1.10)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq126_HTML.gif denote the space of all compact operators from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq127_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq128_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq129_HTML.gif , it is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq130_HTML.gif .

For two sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq132_HTML.gif of positive numbers, the expression https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq133_HTML.gif means that there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq135_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ11_HTML.gif
(1.11)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq136_HTML.gif denote the space of all compact operators from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq137_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq138_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq139_HTML.gif , it is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq140_HTML.gif .

Now, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq141_HTML.gif denotes the approximation numbers of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq142_HTML.gif (see, e.g., [27, Section 1.16.1]). Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ12_HTML.gif
(1.12)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq144_HTML.gif be two Banach spaces and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq145_HTML.gif continuously and densely embedded into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq147_HTML.gif .

We let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq148_HTML.gif denote the space of all functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq149_HTML.gif possessing generalized derivatives https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq150_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq151_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ13_HTML.gif
(1.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq152_HTML.gif . Consider the following weighted spaces of functions:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ14_HTML.gif
(1.14)

2. Background

The embedding theorems play a key role in the perturbation theory of DOEs. For estimating lower order derivatives, we use following embedding theorems from [24].

Theorem A1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq154_HTML.gif and suppose that the following conditions are satisfied:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq155_HTML.gif is a Banach space satisfying the multiplier condition with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq157_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq158_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq159_HTML.gif -positive operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq160_HTML.gif ,

(3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq162_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq163_HTML.gif -tuples of nonnegative integer such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ15_HTML.gif
(2.1)

(4) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq164_HTML.gif is a region such that there exists a bounded linear extension operator from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq165_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq166_HTML.gif .

Then, the embedding https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq167_HTML.gif is continuous. Moreover, for all positive number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq169_HTML.gif , the following estimate holds
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ16_HTML.gif
(2.2)

Theorem A2.

Suppose that all conditions of Theorem A1 are satisfied. Moreover, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq171_HTML.gif be a bounded region and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq172_HTML.gif . Then, the embedding
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ17_HTML.gif
(2.3)

is compact.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq173_HTML.gif denote the closure of the linear span of the root vectors of the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq174_HTML.gif .

From [18, Theorem  3.4.1], we have the following.

Theorem A3.

Assume that

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq175_HTML.gif is an UMD space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq176_HTML.gif is an operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq178_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq179_HTML.gif are non overlapping, differentiable arcs in the complex plane starting at the origin. Suppose that each of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq180_HTML.gif regions into which the planes are divided by these arcs is contained in an angular sector of opening less then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq181_HTML.gif ,

(3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq182_HTML.gif is an integer so that the resolvent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq183_HTML.gif satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ18_HTML.gif
(2.4)

as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq184_HTML.gif along any of the arcs https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq185_HTML.gif .

Then, the subspace https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq186_HTML.gif contains the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq187_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ19_HTML.gif
(2.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ20_HTML.gif
(2.6)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq188_HTML.gif denote the embedding operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq189_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq190_HTML.gif .

From [15, Theorem 2.8], we have the following.

Theorem A4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq192_HTML.gif be two Banach spaces possessing bases. Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ21_HTML.gif
(2.7)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ22_HTML.gif
(2.8)

3. Statement of the Problem

Consider the BVPs for the degenerate anisotropic DOE
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ23_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ24_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ25_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq195_HTML.gif are complex numbers, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq196_HTML.gif are complex-valued functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq198_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq199_HTML.gif are linear operators in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq200_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq201_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq202_HTML.gif are such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ26_HTML.gif
(3.4)

A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq203_HTML.gif and satisfying (3.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq204_HTML.gif is said to be solution of the problem (3.1)-(3.2).

We say the problem (3.1)-(3.2) is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq205_HTML.gif -separable if for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq206_HTML.gif , there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq207_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq208_HTML.gif of the problem (3.1)-(3.2) and a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq209_HTML.gif depending only https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq210_HTML.gif such that the coercive estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ27_HTML.gif
(3.5)

holds.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq211_HTML.gif be a differential operator generated by problem (3.1)-(3.2) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq212_HTML.gif ; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ28_HTML.gif
(3.6)

We say the problem (3.1)-(3.2) is Fredholm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq213_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq214_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq215_HTML.gif is a conjugate of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq216_HTML.gif .

Remark 3.1.

Under the substitutions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ29_HTML.gif
(3.7)
the spaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq218_HTML.gif are mapped isomorphically onto the weighted spaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq220_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ30_HTML.gif
(3.8)
Moreover, under the substitution (3.7) the problem (3.1)-(3.2) reduces to the nondegenerate BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ31_HTML.gif
(3.9)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ32_HTML.gif
(3.10)
By denoting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq223_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq224_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq225_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq226_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq228_HTML.gif again by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq229_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq231_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq236_HTML.gif , respectively, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ33_HTML.gif
(3.11)

4. BVPs for Partial DOE

Let us first consider the BVP for the anisotropic type DOE with constant coefficients
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ34_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ35_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq237_HTML.gif are boundary conditions defined by (3.2), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq238_HTML.gif are complex numbers, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq239_HTML.gif is a complex parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq240_HTML.gif is a linear operator in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq241_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq242_HTML.gif be the roots of the characteristic equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ36_HTML.gif
(4.3)
Now, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ37_HTML.gif
(4.4)

By applying the trace theorem [27, Section 1.8.2], we have the following.

Theorem A5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq244_HTML.gif be integer numbers, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq247_HTML.gif . Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq248_HTML.gif , the transformations https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq249_HTML.gif are bounded linear from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq250_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq251_HTML.gif , and the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ38_HTML.gif
(4.5)

Proof.

It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ39_HTML.gif
(4.6)

Then, by applying the trace theorem [27, Section 1.8.2] to the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq252_HTML.gif , we obtain the assertion.

Condition 1.

Assume that the following conditions are satisfied:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq253_HTML.gif is a Banach space satisfying the multiplier condition with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq254_HTML.gif and the weight function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq256_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq257_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq258_HTML.gif -positive operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq259_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq260_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq261_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ40_HTML.gif
(4.7)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq263_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq264_HTML.gif denote the operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq265_HTML.gif generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.

Theorem A6.

Let Condition 1 be satisfied. Then,
  1. (a)
    the problem (4.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq266_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq267_HTML.gif with sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq268_HTML.gif has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq269_HTML.gif that belongs to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq270_HTML.gif and the following coercive uniform estimate holds:
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ41_HTML.gif
    (4.8)
     

(b)the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq271_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq272_HTML.gif -positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq273_HTML.gif .

From Theorems A5 and A6 we have.

Theorem A7.

Suppose that Condition 1 is satisfied. Then, for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq274_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq275_HTML.gif the problem (4.1) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq276_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq277_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq278_HTML.gif . Moreover, the following uniform coercive estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ42_HTML.gif
(4.9)
Consider BVP (3.11). Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq279_HTML.gif be roots of the characteristic equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ43_HTML.gif
(4.10)

Condition 2.

Suppose the following conditions are satisfied:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq280_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ44_HTML.gif
(4.11)
for
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ45_HTML.gif
(4.12)

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq281_HTML.gif is a Banach space satisfying the multiplier condition with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq282_HTML.gif and the weighted function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq283_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq284_HTML.gif .

Remark 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq285_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq286_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq287_HTML.gif are real-valued positive functions. Then, Condition 2 is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq288_HTML.gif .

Consider the inhomogenous BVP (3.1)-(3.2); that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ46_HTML.gif
(4.13)

Lemma 4.2.

Assume that Condition 2 is satisfied and the following hold:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq289_HTML.gif is a uniformly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq290_HTML.gif -positive operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq291_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq292_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq293_HTML.gif are continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq294_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq295_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq297_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq298_HTML.gif .

Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq299_HTML.gif and for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq300_HTML.gif the following coercive uniform estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ47_HTML.gif
(4.14)

for the solution of problem (4.13).

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq301_HTML.gif be regions covering https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq302_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq303_HTML.gif be a corresponding partition of unity; that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq304_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq306_HTML.gif . Now, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq307_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq308_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ48_HTML.gif
(4.15)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ49_HTML.gif
(4.16)
here, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq309_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq310_HTML.gif are boundary operators which orders less than https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq311_HTML.gif . Freezing the coefficients of (4.15), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ50_HTML.gif
(4.17)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ51_HTML.gif
(4.18)
It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq312_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq313_HTML.gif on neighborhoods of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq314_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ52_HTML.gif
(4.19)
on neighborhoods of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq315_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq316_HTML.gif on other parts of the domains https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq317_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq318_HTML.gif are positive constants. Hence, the problems (4.17) are generated locally only on parts of the boundary. Then, by Theorem A7 problem (4.17) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq319_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq320_HTML.gif the following coercive estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ53_HTML.gif
(4.20)
From the representation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq321_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq322_HTML.gif and in view of the boundedness of the coefficients, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ54_HTML.gif
(4.21)
Now, applying Theorem A1 and by using the smoothness of the coefficients of (4.16), (4.18) and choosing the diameters of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq323_HTML.gif so small, we see there is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq324_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq325_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ55_HTML.gif
(4.22)
Then, using Theorem A5 and using the smoothness of the coefficients of (4.16), (4.18), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ56_HTML.gif
(4.23)
Now, using Theorem A1, we get that there is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq326_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq327_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ57_HTML.gif
(4.24)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ58_HTML.gif
(4.25)
Using the above estimates, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ59_HTML.gif
(4.26)
Consequently, from (4.22)–(4.26), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ60_HTML.gif
(4.27)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq328_HTML.gif from the above inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ61_HTML.gif
(4.28)

Then, by using the equality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq329_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq330_HTML.gif and the above estimates, we get (4.14).

Condition 3.

Suppose that part (1.1) of Condition 1 is satisfied and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq331_HTML.gif is a Banach space satisfying the multiplier condition with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq332_HTML.gif and the weighted function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq333_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq334_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq335_HTML.gif .

Consider the problem (3.11). Reasoning as in the proof of Lemma 4.2, we obtain.

Proposition 4.3.

Assume Condition 3 hold and suppose that

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq336_HTML.gif is a uniformly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq337_HTML.gif -positive operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq338_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq339_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq340_HTML.gif are continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq341_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq342_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq343_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq344_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq345_HTML.gif .

Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq346_HTML.gif and for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq347_HTML.gif , the following coercive uniform estimate holds
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ62_HTML.gif
(4.29)

for the solution of problem (3.11).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq348_HTML.gif denote the operator generated by problem (3.11) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq349_HTML.gif ; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ63_HTML.gif
(4.30)

Theorem 4.4.

Assume that Condition 3 is satisfied and that the following hold:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq350_HTML.gif is a uniformly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq351_HTML.gif -positive operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq352_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq353_HTML.gif are continuous functions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq354_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq355_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq356_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq357_HTML.gif .

Then, problem (3.11) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq358_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq359_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq360_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq361_HTML.gif with large enough https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq362_HTML.gif . Moreover, the following coercive uniform estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ64_HTML.gif
(4.31)

Proof.

By Proposition 4.3 for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq363_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ65_HTML.gif
(4.32)
It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ66_HTML.gif
(4.33)
Hence, by using the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq364_HTML.gif and applying Theorem A1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ67_HTML.gif
(4.34)
From the above estimate, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ68_HTML.gif
(4.35)
The estimate (4.35) implies that problem (3.11) has a unique solution and that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq365_HTML.gif has a bounded inverse in its rank space. We need to show that this rank space coincides with the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq366_HTML.gif ; that is, we have to show that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq367_HTML.gif , there is a unique solution of the problem (3.11). We consider the smooth functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq368_HTML.gif with respect to a partition of unity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq369_HTML.gif on the region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq370_HTML.gif that equals one on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq371_HTML.gif , where supp https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq372_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq373_HTML.gif . Let us construct for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq374_HTML.gif the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq375_HTML.gif that are defined on the regions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq376_HTML.gif and satisfying problem (3.11). The problem (3.11) can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ69_HTML.gif
(4.36)
Consider operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq377_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq378_HTML.gif that are generated by the BVPs (4.17); that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ70_HTML.gif
(4.37)
By virtue of Theorem A6, the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq379_HTML.gif have inverses https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq380_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq381_HTML.gif and for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq382_HTML.gif . Moreover, the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq383_HTML.gif are bounded from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq384_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq385_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq386_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ71_HTML.gif
(4.38)
Extending https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq387_HTML.gif to zero outside of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq388_HTML.gif in the equalities (4.36), and using the substitutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq389_HTML.gif , we obtain the operator equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ72_HTML.gif
(4.39)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq390_HTML.gif are bounded linear operators in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq391_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ73_HTML.gif
(4.40)
In fact, because of the smoothness of the coefficients of the expression https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq392_HTML.gif and from the estimate (4.38), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq393_HTML.gif with sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq394_HTML.gif , there is a sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq395_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ74_HTML.gif
(4.41)
Moreover, from assumption (2.2) of Theorem 4.4 and Theorem A1 for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq396_HTML.gif , there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq397_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ75_HTML.gif
(4.42)
Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq398_HTML.gif with sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq399_HTML.gif , there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq400_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq401_HTML.gif . Consequently, (4.39) for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq402_HTML.gif have a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq403_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ76_HTML.gif
(4.43)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq404_HTML.gif are bounded linear operators from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq405_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq406_HTML.gif . Thus, the functions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ77_HTML.gif
(4.44)
are solutions of (4.38). Consider the following linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq407_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq408_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ78_HTML.gif
(4.45)
It is clear from the constructions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq409_HTML.gif and from the estimate (4.39) that the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq410_HTML.gif are bounded linear from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq411_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq412_HTML.gif , and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq413_HTML.gif with sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq414_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ79_HTML.gif
(4.46)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq415_HTML.gif is a bounded linear operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq416_HTML.gif . Since the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq417_HTML.gif coincide with the inverse of the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq418_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq419_HTML.gif , then acting on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq420_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq421_HTML.gif gives
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ80_HTML.gif
(4.47)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq422_HTML.gif are bounded linear operators defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ81_HTML.gif
(4.48)
Indeed, from Theorem A1 and estimate (4.46) and from the expression https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq423_HTML.gif , we obtain that the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq424_HTML.gif are bounded linear from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq425_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq426_HTML.gif , and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq427_HTML.gif with sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq428_HTML.gif , there is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq429_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq430_HTML.gif . Therefore, there exists a bounded linear invertible operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq431_HTML.gif ; that is, we infer for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq432_HTML.gif that the BVP (3.11) has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ82_HTML.gif
(4.49)

Result 1.

Theorem 4.4 implies that the resolvent https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq433_HTML.gif satisfies the following anisotropic type sharp estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ83_HTML.gif
(4.50)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq434_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq435_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq436_HTML.gif denote the operator generated by BVP (3.1)-(3.2). From Theorem 4.4 and Remark 3.1, we get the following.

Result 2.

Assume all the conditions of Theorem 4.4 hold. Then,

(a)the problem (3.1)-(3.2) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq437_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq438_HTML.gif and for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq439_HTML.gif has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq440_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq441_HTML.gif , and the following coercive uniform estimate holds
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ84_HTML.gif
(4.51)

(b)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq442_HTML.gif , then the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq443_HTML.gif is Fredholm from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq444_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq445_HTML.gif .

Example 4.5.

Now, let us consider a special case of (3.1)-(3.2). Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq446_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq447_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq448_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq449_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq450_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq451_HTML.gif ; that is, consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ85_HTML.gif
(4.52)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ86_HTML.gif
(4.53)
Theorem 4.4 implies that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq452_HTML.gif , problem (4.52) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq453_HTML.gif satisfying the following coercive estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ87_HTML.gif
(4.54)

Example 4.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq454_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq455_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq456_HTML.gif are positive continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq457_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq458_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq459_HTML.gif is a diagonal matrix-function with continuous components https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq460_HTML.gif .

Then, we obtain the separability of the following BVPs for the system of anisotropic PDEs with varying coefficients:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ88_HTML.gif
(4.55)

in the vector-valued space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq461_HTML.gif .

5. The Spectral Properties of Anisotropic Differential Operators

Consider the following degenerated BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ89_HTML.gif
(5.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ90_HTML.gif
(5.2)

Consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq462_HTML.gif generated by problem (5.1).

Theorem 5.1.

Let all the conditions of Theorem 4.4 hold for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq463_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq464_HTML.gif . Then, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq465_HTML.gif is Fredholm from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq466_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq467_HTML.gif .

Proof.

Theorem 4.4 implies that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq468_HTML.gif for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq469_HTML.gif has a bounded inverse https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq470_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq471_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq472_HTML.gif ; that is, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq473_HTML.gif is Fredholm from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq474_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq475_HTML.gif . Then, from Theorem A2 and the perturbation theory of linear operators, we obtain that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq476_HTML.gif is Fredholm from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq477_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq478_HTML.gif .

Theorem 5.2.

Suppose that all the conditions of Theorem 5.1 are satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq479_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq480_HTML.gif is a Banach space with a basis and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ91_HTML.gif
(5.3)

Then,

(a)for a sufficiently large positive https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq481_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ92_HTML.gif
(5.4)

(b)the system of root functions of the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq482_HTML.gif is complete in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq483_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq484_HTML.gif denote the embedding operator from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq485_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq486_HTML.gif . From Result 2, there exists a resolvent operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq487_HTML.gif which is bounded from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq488_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq489_HTML.gif . Moreover, from Theorem A4 and Remark 3.1, we get that the embedding operator
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ93_HTML.gif
(5.5)
is compact and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ94_HTML.gif
(5.6)
It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ95_HTML.gif
(5.7)
Hence, from relations (5.6) and (5.7), we obtain (5.4). Now, Result 1 implies that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq490_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq491_HTML.gif is positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq492_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ96_HTML.gif
(5.8)

Then, from (4.52) and (5.6), we obtain assertion (b).

Consider now the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq493_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq494_HTML.gif generated by the nondegenerate BVP obtained from (5.1) under the mapping (3.7); that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ97_HTML.gif
(5.9)

From Theorem 5.2 and Remark 3.1, we get the following.

Result 3.

Let all the conditions of Theorem 5.1 hold. Then, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq495_HTML.gif is Fredholm from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq496_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq497_HTML.gif .

Result 4.

Then,

(a)for a sufficiently large positive https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq498_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ98_HTML.gif
(5.10)

(b)the system of root functions of the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq499_HTML.gif is complete in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq500_HTML.gif .

6. BVPs for Degenerate Quasielliptic PDE

In this section, maximal regularity properties of degenerate anisotropic differential equations are studied. Maximal regularity properties for PDEs have been studied, for example, in [3] for smooth domains and in [28] for nonsmooth domains.

Consider the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ99_HTML.gif
(6.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq501_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq502_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq503_HTML.gif are complex number, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq504_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ100_HTML.gif
(6.2)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq505_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq506_HTML.gif . Now, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq507_HTML.gif will denote the space of all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq508_HTML.gif -summable scalar-valued functions with mixed norm (see, e.g., [29, Section 1, page 6]), that is, the space of all measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq509_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq510_HTML.gif , for which
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ101_HTML.gif
(6.3)

Analogously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq511_HTML.gif denotes the Sobolev space with corresponding mixed norm.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq512_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq513_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq514_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq515_HTML.gif denote the roots of the equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ102_HTML.gif
(6.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq516_HTML.gif denote the operator generated by BVP (6.1). Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ103_HTML.gif
(6.5)

Theorem 6.1.

Let the following conditions be satisfied:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq517_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq518_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq519_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq520_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq521_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq522_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq523_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq524_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq525_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq526_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq527_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq528_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq529_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq530_HTML.gif ,

(3)for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq531_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq532_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq533_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq534_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq535_HTML.gif let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ104_HTML.gif
(6.6)
(4)for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq536_HTML.gif , the local BVPs in local coordinates corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq537_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ105_HTML.gif
(6.7)

has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq538_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq539_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq540_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq541_HTML.gif ,

(5) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq542_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq543_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ106_HTML.gif
(6.8)

Then,

(a)the following coercive estimate
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ107_HTML.gif
(6.9)

holds for the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq544_HTML.gif of problem (6.1),

(b)for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq545_HTML.gif and for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq546_HTML.gif , there exists a resolvent https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq547_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ108_HTML.gif
(6.10)

(c)the problem (6.1) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq548_HTML.gif is Fredholm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq549_HTML.gif ,

(d)the relation with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq550_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ109_HTML.gif
(6.11)

holds,

(e)for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq551_HTML.gif the system of root functions of the BVP (6.1) is complete in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq552_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq553_HTML.gif . Then, from [3, Theorem 3.6], part (1.1) of Condition 1 is satisfied. Consider the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq554_HTML.gif which is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ110_HTML.gif
(6.12)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq555_HTML.gif , we also consider operators
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ111_HTML.gif
(6.13)
The problem (6.1) can be rewritten as the form of (3.1)-(3.2), where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq556_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq557_HTML.gif are functions with values in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq558_HTML.gif . From [3, Theorem 8.2] problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ112_HTML.gif
(6.14)
has a unique solution for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq559_HTML.gif and arg https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq560_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq561_HTML.gif . Moreover, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq562_HTML.gif , generated by (5.8) is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq563_HTML.gif -positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq564_HTML.gif ; that is, part (2.2) of Condition 1 holds. From (2.2), (3.7), and by [29, Section 18], we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ113_HTML.gif
(6.15)
that is, all the conditions of Theorem 5.2 and Result 4 are fulfilled. As a result, we obtain assertion (a) and (b) of the theorem. Also, it is known (e.g., [27, Theorem 3.2.5, Section 4.10]) that the embedding https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq565_HTML.gif is compact and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ114_HTML.gif
(6.16)

Then, Results 3 and 4 imply assertions (c), (d), (e).

7. Boundary Value Problems for Infinite Systems of Degenerate PDE

Consider the infinity systems of BVP for the degenerate anisotropic PDE
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ115_HTML.gif
(7.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq566_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq567_HTML.gif are complex-valued functions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq568_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq569_HTML.gif are complex numbers. Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ116_HTML.gif
(7.2)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq570_HTML.gif denote the operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq571_HTML.gif generated by problem (7.1). Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ117_HTML.gif
(7.3)

Theorem 7.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq572_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq573_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq574_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq575_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq576_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq577_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq578_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq579_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq580_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq581_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq582_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq583_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq584_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq585_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ118_HTML.gif
(7.4)

Then,

(a)for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq586_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq587_HTML.gif and sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq588_HTML.gif , the problem (7.1) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq589_HTML.gif that belongs to the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq590_HTML.gif and the following coercive estimate holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ119_HTML.gif
(7.5)
(b)there exists a resolvent https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq591_HTML.gif of the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq592_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ120_HTML.gif
(7.6)

(c)for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq593_HTML.gif , the system of root functions of the BVP (7.1) is complete in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq594_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq595_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq596_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq597_HTML.gif be infinite matrices such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ121_HTML.gif
(7.7)
It is clear that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq598_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq599_HTML.gif -positive in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq600_HTML.gif . The problem (7.1) can be rewritten in the form (1.1). From Theorem 4.4, we obtain that problem (7.1) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq601_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq602_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_IEq603_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F268032/MediaObjects/13661_2010_Article_32_Equ122_HTML.gif
(7.8)

From the above estimate, we obtain assertions (a) and (b). The assertion (c) is obtained from Result 4.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Department of Mathematics, National University of Ireland
(3)
Department of Electronics Engineering and Communication, Okan University, Akfirat, Tuzla

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© Ravi Agarwal et al. 2011

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