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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 237657, 12 pages
http://dx.doi.org/10.1155/2012/237657
Research Article

Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in HΓΆlder Spaces

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

Received 30 March 2012; Accepted 17 April 2012

Academic Editor: AllaberenΒ Ashyralyev

Copyright Β© 2012 Okan Gercek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A first order of accuracy difference scheme for the approximate solution of abstract nonlocal boundary value problem βˆ’π‘‘2𝑒(𝑑)/𝑑𝑑2+sign(𝑑)𝐴𝑒(𝑑)=𝑔(𝑑), (0≀𝑑≀1), 𝑑𝑒(𝑑)/𝑑𝑑+sign(𝑑)𝐴𝑒(𝑑)=𝑓(𝑑), (βˆ’1≀𝑑≀0), 𝑒(0+)=𝑒(0βˆ’),π‘’ξ…ž(0+)=π‘’ξ…ž(0βˆ’),and𝑒(1)=𝑒(βˆ’1)+πœ‡ for differential equations in a Hilbert space 𝐻 with a self-adjoint positive definite operator A is considered. The well-posedness of this difference scheme in HΓΆlder spaces without a weight is established. Moreover, as applications, coercivity estimates in HΓΆlder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.

1. Introduction

Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function (see, e.g., [1–15] and the references therein).

Well-posedness of difference schemes of elliptic-parabolic equations with nonlocal boundary conditions in HΓΆlder spaces with a weight was studied in [16–19].

In paper [20], the well-posedness of abstract nonlocal boundary value problem βˆ’π‘‘2𝑒(𝑑)𝑑𝑑2+sign(𝑑)𝐴𝑒(𝑑)=𝑔(𝑑),(0≀𝑑≀1),𝑑𝑒(𝑑)𝑑𝑑+sign(𝑑)𝐴𝑒(𝑑)=𝑓(𝑑),(βˆ’1≀𝑑≀0),𝑒(0+)=𝑒(0βˆ’),π‘’ξ…ž(0+)=π‘’ξ…ž(0βˆ’),𝑒(1)=𝑒(βˆ’1)+πœ‡(1.1) in HΓΆlder spaces without a weight was established. The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations were obtained.

In the present paper, the first order of accuracy difference scheme βˆ’πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έ+π΄π‘’π‘˜=π‘”π‘˜,π‘”π‘˜ξ€·π‘‘=π‘”π‘˜ξ€Έ,π‘‘π‘˜πœ=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,βˆ’1ξ€·π‘’π‘˜βˆ’π‘’π‘˜βˆ’1ξ€Έβˆ’π΄π‘’π‘˜βˆ’1=π‘“π‘˜,π‘“π‘˜ξ€·π‘‘=π‘“π‘˜βˆ’1ξ€Έ,π‘‘π‘˜βˆ’1𝑒=(π‘˜βˆ’1)𝜏,βˆ’π‘+1β‰€π‘˜β‰€βˆ’1,𝑁=π‘’βˆ’π‘+πœ‡,𝑒1βˆ’π‘’0=𝑒0βˆ’π‘’βˆ’1(1.2) for the approximate solution of problem (1.1) is considered. The well-posedness of difference scheme (1.2) in HΓΆlder spaces without a weight is established. As an application, coercivity inequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.

Throughout the paper, 𝐻 denotes a Hilbert space and 𝐴 is a self-adjoint positive definite operator with 𝐴β‰₯𝛿𝐼 for some 𝛿>𝛿0>0. Then, it is wellknown that √𝐡=(1/2)(𝜏𝐴+𝐴(4+𝜏2𝐴)) is a self-adjoint positive definite operator and 𝐡β‰₯𝛿1/2𝐼. Furthermore, 𝑅=(𝐼+𝜏𝐡)βˆ’1 and 𝑃=𝑃(𝜏𝐴)=(𝐼+𝜏𝐴)βˆ’1 which are defined on the whole space 𝐻, are bounded operators, where 𝐼 is the identity operator.

2. Well-Posedness of (1.2)

First of all, let us start with some auxiliary lemmas that are used throughout the paper.

Lemma 2.1. The following estimates are satisfied [19, 21, 22]: β€–β€–π‘ƒπ‘˜β€–β€–π»β†’π»β‰€π‘€(𝛿)(1+π›Ώπœ)βˆ’π‘˜β€–β€–,π‘˜πœπ΄π‘ƒπ‘˜β€–β€–π»β†’π»β€–β€–π‘…β‰€π‘€(𝛿),π‘˜β€–β€–π»β†’π»β‰€π‘€(𝛿)(1+π›Ώπœ)βˆ’π‘˜β€–β€–,π‘˜πœπ΅π‘…π‘˜β€–β€–π»β†’π»β€–β€–π‘ƒβ‰€π‘€(𝛿),π‘˜βˆ’π‘’βˆ’π‘˜πœπ΄β€–β€–π»β†’π»β‰€π‘€(𝛿)π‘˜,β€–β€–π‘…π‘˜βˆ’π‘’βˆ’π‘˜πœπ΄1/2‖‖𝐻→𝐻≀𝑀(𝛿)π‘˜,β€–β€–ξ€·πΌβˆ’π‘…2π‘ξ€Έβˆ’1‖‖𝐻→𝐻≀𝑀(𝛿),π‘˜β‰₯1,𝛿>0,(2.1) for some 𝑀(𝛿)>0, which is independent of 𝜏 is a positive small number.

Let 𝐹𝜏(𝐻)=𝐹([π‘Ž,𝑏]𝜏,𝐻) be the linear space of mesh functions πœ‘πœ={πœ‘π‘˜}π‘π‘π‘π‘Ž defined on [π‘Ž,𝑏]𝜏={π‘‘π‘˜=π‘˜β„Ž,π‘π‘Žβ‰€π‘˜β‰€π‘π‘,π‘π‘Žπœ=π‘Ž,π‘π‘πœ=𝑏} with values in the Hilbert space 𝐻. Next, 𝐢([π‘Ž,𝑏]𝜏,𝐻), 𝐢𝛼([βˆ’1,1]𝜏,𝐻), 𝐢𝛼/2([βˆ’1,0]𝜏,𝐻), and 𝐢𝛼([0,1]𝜏,𝐻)(0<𝛼<1) denote Banach spaces on 𝐹𝜏(𝐻) with norms: β€–πœ‘πœβ€–πΆ([π‘Ž,𝑏]𝜏,𝐻)=maxπ‘π‘Žβ‰€π‘˜β‰€π‘π‘β€–β€–πœ‘π‘˜β€–β€–π»,β€–πœ‘πœβ€–πΆπ›Ό([βˆ’1,1]𝜏,𝐻)=β€–πœ‘πœβ€–πΆ([βˆ’1,1]𝜏,𝐻)+supβˆ’π‘β‰€π‘˜<π‘˜+π‘Ÿβ‰€0β€–β€–πœ‘π‘˜+π‘Ÿβˆ’πœ‘π‘˜β€–β€–π»π‘Ÿβˆ’π›Ό/2+sup1β‰€π‘˜<π‘˜+π‘Ÿβ‰€π‘βˆ’1β€–β€–πœ‘π‘˜+π‘Ÿβˆ’πœ‘π‘˜β€–β€–π»π‘Ÿβˆ’π›Ό,β€–πœ‘πœβ€–πΆπ›Ό/2([βˆ’1,0]𝜏,𝐻)=β€–πœ‘πœβ€–πΆ([βˆ’1,0]𝜏,𝐻)+supβˆ’π‘β‰€π‘˜<π‘˜+π‘Ÿβ‰€0β€–β€–πœ‘π‘˜+π‘Ÿβˆ’πœ‘π‘˜β€–β€–π»π‘Ÿβˆ’π›Ό/2,β€–πœ‘πœβ€–πΆπ›Ό([0,1]𝜏,𝐻)=β€–πœ‘πœβ€–πΆ([0,1]𝜏,𝐻)+sup1β‰€π‘˜<π‘˜+π‘Ÿβ‰€π‘βˆ’1β€–β€–πœ‘π‘˜+π‘Ÿβˆ’πœ‘π‘˜β€–β€–π»π‘Ÿβˆ’π›Ό.(2.2)

With the help of the self-adjoint positive definite operator 𝐡 in a Hilbert space 𝐻, the Banach space 𝐸𝛼=𝐸𝛼(𝐡,𝐻)(0<𝛼<1) consists of those π‘£βˆˆπ» for which the norm (see [22, 23]): ‖𝑣‖𝐸𝛼=sup𝑧>0𝑧𝛼‖‖𝐡(𝑧+𝐡)βˆ’1𝑣‖‖𝐻+‖𝑣‖𝐻,(2.3) is finite. By the definition of 𝐸𝛼(𝐡,𝐻), 𝐷(𝐡)βŠ‚πΈπ›Ό(𝐡,𝐻)βŠ‚πΈπ›½(𝐡,𝐻)βŠ‚π»,(2.4) for all 𝛽<𝛼.

Lemma 2.2. For 0<𝛼<1, the norms of the spaces 𝐸𝛼(𝐡,𝐻) and 𝐸𝛼/2(𝐴,𝐻) are equivalent (see [24]).

Theorem 2.3. Suppose πœ‡βˆˆπ·(𝐴),π΄πœ‡βˆˆπΈπ›Ό(𝐡,𝐻),𝑓0+𝑔0βˆˆπΈπ›Ό/2(𝐴,𝐻),π‘“βˆ’π‘+π‘”π‘βˆˆπΈπ›Ό(𝐡,𝐻), 𝑔(𝑑)βˆˆπΆπ›Ό([0,1]𝜏,𝐻), and 𝑓(𝑑)βˆˆπΆπ›Ό/2([βˆ’1,0]𝜏,𝐻),0<𝛼<1. Boundary value problem (1.2) is wellposed in HΓΆlder space 𝐢𝛼([βˆ’1,1]𝜏,𝐻) and the following coercivity inequality holds: β€–β€–ξ€½πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έξ€Ύ1π‘βˆ’1‖‖𝐢𝛼([0,1]𝜏,𝐻)+β€–β€–ξ€½π΄π‘’π‘˜ξ€Ύπ‘βˆ’1βˆ’π‘β€–β€–πΆπ›Ό([βˆ’1,1]𝜏,𝐻)+β€–β€–ξ€½πœβˆ’1ξ€·π‘’π‘˜βˆ’π‘’π‘˜βˆ’1ξ€Έξ€Ύ0βˆ’π‘+1‖‖𝐢𝛼/2([βˆ’1,0]𝜏,𝐻)ξ‚Έβ‰€π‘€β€–π΄πœ‡β€–πΈπ›Ό(𝐡,𝐻)+1𝛼(1βˆ’π›Ό)β€–π‘“πœβ€–πΆπ›Ό/2([βˆ’1,0]𝜏,𝐻)+β€–π‘”πœβ€–πΆπ›Ό([0,1]𝜏,𝐻)ξ€»+‖‖𝑓(𝐼+𝜏𝐡)0+𝑔0‖‖𝐸𝛼/2(𝐴,𝐻)+‖‖𝑓(𝐼+𝜏𝐡)βˆ’π‘+𝑔𝑁‖‖𝐸𝛼(𝐡,𝐻)ξ‚Ή,(2.5) where 𝑀 is independent of not only π‘“πœ, π‘”πœ, and   πœ‡ but also of 𝜏 and 𝛼.

Proof. First of all, let us get the formulae for solution of problem (1.2). By [21, 25], π‘’π‘˜=ξ€·πΌβˆ’π‘…2π‘ξ€Έβˆ’1ξƒ―ξ€Ίπ‘…π‘˜βˆ’π‘…2π‘βˆ’π‘˜ξ€»ξ€Ίπ‘…πœ‰+π‘βˆ’π‘˜βˆ’π‘…π‘+π‘˜ξ€»πœ“βˆ’ξ€Ίπ‘…π‘βˆ’π‘˜βˆ’π‘…π‘+π‘˜ξ€»(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1ξ€Ίπ‘…π‘βˆ’π‘ βˆ’π‘…π‘+π‘ ξ€»π‘”π‘ πœξƒ°+(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1𝑅|π‘˜βˆ’π‘ |βˆ’π‘…π‘˜+π‘ ξ€»π‘”π‘ πœ,1β‰€π‘˜β‰€π‘(2.6) is the solution of boundary value difference problem: βˆ’πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έ+π΄π‘’π‘˜=π‘”π‘˜π‘”π‘˜ξ€·π‘‘=π‘”π‘˜ξ€Έ,π‘‘π‘˜=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,𝑒0=πœ‰,𝑒𝑁𝑒=πœ“,(2.7)π‘˜=π‘ƒβˆ’π‘˜πœ‰βˆ’πœ0𝑠=π‘˜+1π‘ƒπ‘ βˆ’π‘˜π‘“π‘ ,βˆ’π‘β‰€π‘˜β‰€βˆ’1(2.8) is the solution of inverse Cauchy problem: πœβˆ’1ξ€·π‘’π‘˜βˆ’π‘’π‘˜βˆ’1ξ€Έβˆ’π΄π‘’π‘˜βˆ’1=π‘“π‘˜,π‘“π‘˜ξ€·π‘‘=π‘“π‘˜βˆ’1ξ€Έ,π‘‘π‘˜βˆ’1=(π‘˜βˆ’1)𝜏,βˆ’(π‘βˆ’1)β‰€π‘˜β‰€0,𝑒0=πœ‰.(2.9)
Combining the conditions πœ“=π‘’βˆ’π‘+πœ‡,πœ‰=𝑒0 and formulas (2.6), (2.8), we get formulas π‘’π‘˜=ξ€·πΌβˆ’π‘…2π‘ξ€Έβˆ’1ξƒ―ξ€Ίπ‘…π‘˜βˆ’π‘…2π‘βˆ’π‘˜ξ€»π‘’0+ξ€Ίπ‘…π‘βˆ’π‘˜βˆ’π‘…π‘+π‘˜ξ€»ξƒ¬π‘ƒπ‘π‘’0βˆ’πœ0𝑠=βˆ’π‘+1𝑃𝑠+π‘π‘“π‘ ξƒ­βˆ’ξ€Ίπ‘…+πœ‡π‘βˆ’π‘˜βˆ’π‘…π‘+π‘˜ξ€»(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1ξ€Ίπ‘…π‘βˆ’π‘ βˆ’π‘…π‘+π‘ ξ€»π‘”π‘ πœξƒ°+(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1𝑅|π‘˜βˆ’π‘ |βˆ’π‘…π‘˜+π‘ ξ€»π‘”π‘ π‘’πœ,1β‰€π‘˜β‰€π‘,(2.10)π‘˜=π‘ƒβˆ’π‘˜π‘’0βˆ’πœ0𝑠=π‘˜+1π‘ƒπ‘ βˆ’π‘˜π‘“π‘ ,βˆ’π‘β‰€π‘˜β‰€βˆ’1.(2.11)
Operator equation 2𝑒0βˆ’π‘ƒπ‘’0+πœπ‘ƒπ‘“0=ξ€·πΌβˆ’π‘…2π‘ξ€Έβˆ’1ξƒ―ξ€Ίπ‘…βˆ’π‘…2π‘βˆ’1𝑒0+ξ€Ίπ‘…π‘βˆ’1βˆ’π‘…π‘+1×𝑃𝑁𝑒0βˆ’πœ0𝑠=βˆ’π‘+1𝑃𝑠+π‘π‘“π‘ ξƒ­βˆ’ξ€Ίπ‘…+πœ‡π‘βˆ’1βˆ’π‘…π‘+1ξ€»(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1ξ€Ίπ‘…π‘βˆ’π‘ βˆ’π‘…π‘+π‘ ξ€»π‘”π‘ πœξƒ°+(𝐼+𝜏𝐡)(2𝐼+𝜏𝐡)βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1ξ€Ίπ‘…π‘ βˆ’1βˆ’π‘…1+π‘ ξ€»π‘”π‘ πœ(2.12) follows from formulas (2.10), (2.11), and the condition 𝑒1βˆ’π‘’0=𝑒0βˆ’π‘’βˆ’1. As the operator 𝐼+(𝐼+𝜏𝐴)(𝐼+2𝜏𝐴)βˆ’1𝑅2π‘βˆ’1+π΅βˆ’1𝐴(𝐼+2𝜏𝐴)βˆ’1ξ€·πΌβˆ’π‘…2π‘βˆ’1ξ€Έβˆ’(2𝐼+𝜏𝐡)(𝐼+2𝜏𝐴)βˆ’1π‘…π‘π‘ƒπ‘βˆ’1(2.13) has an inverse π‘‡πœ=𝐼+(𝐼+𝜏𝐴)(𝐼+2𝜏𝐴)βˆ’1𝑅2π‘βˆ’1+π΅βˆ’1𝐴(𝐼+2𝜏𝐴)βˆ’1ξ€·πΌβˆ’π‘…2π‘βˆ’1ξ€Έβˆ’(2𝐼+𝜏𝐡)(𝐼+2𝜏𝐴)βˆ’1π‘…π‘π‘ƒπ‘βˆ’1ξ€Έβˆ’1,(2.14) it follows that 𝑒0=π‘‡πœ(𝐼+2𝜏𝐴)βˆ’1(𝐼+𝜏𝐴)ξƒ―ξƒ―(2+𝜏𝐡)π‘…π‘ξƒ¬βˆ’πœ0𝑠=βˆ’π‘+1𝑃𝑠+𝑁𝑓𝑠+πœ‡βˆ’π‘…π‘βˆ’1π΅βˆ’1π‘βˆ’1𝑠=1ξ€Ίπ‘…π‘βˆ’π‘ βˆ’π‘…π‘+π‘ ξ€»π‘”π‘ πœξƒ°+ξ€·πΌβˆ’π‘…2π‘ξ€Έπ΅βˆ’1π‘βˆ’1𝑠=1π‘…π‘ βˆ’1π‘”π‘ ξ€·πœβˆ’πΌβˆ’π‘…2𝑁(𝐼+𝜏𝐡)π΅βˆ’1𝑃𝑓0ξƒ°(2.15) for the solution of operator equation (2.12). Hence, we have formulas (2.10), (2.11), and (2.15) for the solution of difference problem (1.2).
Using formulae (2.10) and (2.15), we can get 𝐴𝑒0=π‘‡πœ(𝐼+2𝜏𝐴)βˆ’1Γ—(𝐼+𝜏𝐴)ξƒ―ξƒ―(2+𝜏𝐡)π‘…π‘ξƒ¬βˆ’πœ0𝑠=βˆ’π‘+1𝐴𝑃𝑠+π‘ξ€·π‘“π‘ βˆ’π‘“βˆ’π‘+1ξ€Έξƒ­+π΄πœ‡βˆ’π‘…π‘βˆ’1π΄π΅βˆ’2ξƒ―π‘βˆ’1𝑠=1π΅π‘…π‘βˆ’π‘ ξ€·π‘”π‘ βˆ’π‘”π‘βˆ’1ξ€Έπœ+π‘βˆ’1𝑠=1𝐡𝑅𝑁+𝑠𝑔1βˆ’π‘”π‘ ξ€Έπœ+ξ€·ξƒ°ξƒ°πΌβˆ’π‘…2π‘ξ€Έπ΄π΅βˆ’2π‘βˆ’1𝑠=1π΅π‘…π‘ βˆ’1ξ€·π‘”π‘ βˆ’π‘”1ξ€Έπœξƒ°+π‘‡πœ(𝐼+2𝜏𝐴)βˆ’1Γ—(𝐼+𝜏𝐴)ξ€½ξ€½(2+𝜏𝐡)π‘…π‘ξ€·π‘ƒπ‘ξ€Έπ‘“βˆ’πΌβˆ’π‘+1βˆ’π‘…π‘βˆ’1π΄π΅βˆ’2ξ€½ξ€·πΌβˆ’π‘…π‘βˆ’1ξ€Έπ‘”π‘βˆ’1βˆ’ξ€·π‘…π‘βˆ’2βˆ’π‘…2π‘βˆ’1𝑔1+ξ€·ξ€Ύξ€ΎπΌβˆ’π‘…2π‘ξ€Έπ΄π΅βˆ’2ξ€·πΌβˆ’π‘…π‘βˆ’1𝑔1βˆ’ξ€·πΌβˆ’π‘…2𝑁(𝐼+𝜏𝐡)π΅βˆ’1𝐴𝑃𝑓0ξ€Ύ,(2.16)𝐴𝑒𝑁=π‘ƒπ‘ξƒ―π‘‡πœ(𝐼+2𝜏𝐴)βˆ’1(×𝐼+𝜏𝐴)ξƒ―ξƒ―(2+𝜏𝐡)π‘…π‘ξƒ¬βˆ’πœ0𝑠=βˆ’π‘+1𝐴𝑃𝑠+π‘ξ€·π‘“π‘ βˆ’π‘“βˆ’π‘+1ξ€Έξƒ­+π΄πœ‡βˆ’π‘…π‘βˆ’1π΄π΅βˆ’2ξƒ―π‘βˆ’1𝑠=1π΅π‘…π‘βˆ’π‘ ξ€·π‘”π‘ βˆ’π‘”π‘βˆ’1ξ€Έπœ+π‘βˆ’1𝑠=1𝐡𝑅𝑁+𝑠𝑔1βˆ’π‘”π‘ ξ€Έπœ+ξ€·ξƒ°ξƒ°πΌβˆ’π‘…2π‘ξ€Έπ΄π΅βˆ’2π‘βˆ’1𝑠=1π΅π‘…π‘ βˆ’1ξ€·π‘”π‘ βˆ’π‘”1ξ€Έπœξƒ°ξƒ°βˆ’πœ0𝑠=βˆ’π‘+1𝐴𝑃𝑠+π‘ξ€·π‘“π‘ βˆ’π‘“βˆ’π‘+1𝑃+π΄πœ‡+π‘ξ€Έπ‘“βˆ’πΌβˆ’π‘+1+π‘ƒπ‘ξ€½π‘‡πœ(𝐼+2𝜏𝐴)βˆ’1Γ—((𝐼+𝜏𝐴)ξ€½ξ€½2+𝜏𝐡)π‘…π‘ξ€·π‘ƒπ‘ξ€Έπ‘“βˆ’πΌβˆ’π‘+1βˆ’π‘…π‘βˆ’1π΄π΅βˆ’2ξ€½ξ€·πΌβˆ’π‘…π‘βˆ’1ξ€Έπ‘”π‘βˆ’1βˆ’ξ€·π‘…π‘βˆ’2βˆ’π‘…2π‘βˆ’1𝑔1+ξ€·ξ€Ύξ€ΎπΌβˆ’π‘…2π‘ξ€Έπ΄π΅βˆ’2ξ€·πΌβˆ’π‘…π‘βˆ’1𝑔1βˆ’ξ€·πΌβˆ’π‘…2𝑁(𝐼+𝜏𝐡)π΅βˆ’1𝐴𝑃𝑓0.ξ€Ύξ€Ύ(2.17)
Finally, we will get coercivity estimate (2.5). It is based on estimates β€–β€–ξ€½πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έξ€Ύ1π‘βˆ’1‖‖𝐢𝛼([0,1]𝜏,𝐻)+β€–β€–ξ€½π΄π‘’π‘˜ξ€Ύ1π‘βˆ’1‖‖𝐢𝛼([0,1]𝜏,𝐻)ξ‚Έ1≀𝑀𝛼(1βˆ’π›Ό)β€–π‘”πœβ€–πΆπ›Ό([0,1]𝜏,𝐻)+‖‖𝐴𝑒0βˆ’π‘”0‖‖𝐸𝛼(𝐡,𝐻)+β€–β€–π΄π‘’π‘βˆ’π‘”π‘β€–β€–πΈπ›Ό(𝐡,𝐻)ξ‚Ή(2.18) for the solution of boundary value difference problem (2.7), β€–β€–ξ€½πœβˆ’1ξ€·π‘’π‘˜βˆ’π‘’π‘˜βˆ’1ξ€Έξ€Ύ0βˆ’π‘+1‖‖𝐢𝛼/2([βˆ’1,0]𝜏,𝐻)+β€–β€–ξ€½π΄π‘’π‘˜ξ€Ύ0βˆ’π‘β€–β€–πΆπ›Ό/2([βˆ’1,0]𝜏,𝐻)ξ‚Έ1≀𝑀(𝛼/2)(1βˆ’π›Ό/2)β€–π‘“πœβ€–πΆπ›Ό/2([βˆ’1,0]𝜏,𝐻)+‖‖𝐴𝑒0+𝑓0‖‖𝐸𝛼(𝐴,𝐻)ξ‚Ή,(2.19) for the solution of inverse Cauchy difference problem (2.9), and ‖‖𝐴𝑒0+𝑓0‖‖𝐸𝛼/2(𝐴,𝐻)≀𝑀𝛼(1βˆ’π›Ό)‖𝑔‖𝐢𝛼([0,1],𝐻)+‖𝑓‖𝐢𝛼/2([βˆ’1,0],𝐻)+π‘€β€–π΄πœ‡β€–πΈπ›Ό(𝐡,𝐻)+‖‖𝑓0+𝑔0‖‖𝐸𝛼/2(𝐴,𝐻)ξ‚„,‖‖𝐴𝑒0βˆ’π‘”0‖‖𝐸𝛼(𝐡,𝐻)≀𝑀𝛼(1βˆ’π›Ό)‖𝑓‖𝐢𝛼/2([βˆ’1,0],𝐻)+‖𝑔‖𝐢𝛼([0,1],𝐻)+π‘€β€–π΄πœ‡β€–πΈπ›Ό(𝐡,𝐻)+‖‖𝑓0+𝑔0‖‖𝐸𝛼/2(𝐴,𝐻)ξ‚„,β€–β€–π΄π‘’π‘βˆ’π‘”π‘β€–β€–πΈπ›Ό(𝐡,𝐻)≀𝑀‖𝛼(1βˆ’π›Ό)𝑓‖𝐢𝛼/2([βˆ’1,0],𝐻)+‖𝑔‖𝐢𝛼([0,1],𝐻)+π‘€β€–π΄πœ‡β€–πΈπ›Ό(𝐡,𝐻)+‖‖𝑓0+𝑔0‖‖𝐸𝛼/2(𝐴,𝐻)+β€–β€–π‘“βˆ’π‘+𝑔𝑁‖‖𝐸𝛼(𝐡,𝐻)ξ‚„(2.20) for the solution of problem (1.2). Estimates (2.18) and (2.19) were established in [21, 25], respectively.
Estimates (2.20) are derived from the formulas (2.16) and (2.17) for the solution of problem (1.2), estimates (2.1) and following estimates β€–β€–π‘…π‘˜β€–β€–(𝜏𝐡)𝐻→𝐻‖‖≀𝑀,1β‰€π‘˜β‰€π‘,πΌβˆ’π‘…2𝑁(𝜏𝐡)βˆ’1‖‖𝐻→𝐻‖‖𝑅≀𝑀,π‘˜β‰₯1,π‘˜+π‘Ÿ(𝜏𝐡)βˆ’π‘…π‘˜β€–β€–(𝜏𝐡)𝐻→𝐻≀𝑀(π‘Ÿ)𝛼(π‘˜+π‘Ÿ)𝛼‖‖,1β‰€π‘˜<π‘˜+π‘Ÿβ‰€π‘,0≀𝛼≀1,(πΌβˆ’π‘…(𝜏𝐡))2(𝜏𝐡)βˆ’2‖‖𝐻→𝐻‖‖≀𝑀,(𝐼+𝑅(𝜏𝐡))βˆ’1‖‖𝐻→𝐻‖‖𝑇≀𝑀,πœβ€–β€–π»β†’π»β€–β€–β‰€π‘€,π΅π‘…π‘ƒπ‘‡πœβ€–β€–π»β†’π»β‰€π‘€,(2.21) which were established in [26]. This finalizes the proof of Theorem 2.3.

3. An Application

In this section, an application of the abstract Theorem 2.3 is considered. First, let Ξ© be the unit open cube in the 𝑛-dimensional Euclidean space ℝ𝑛(0<π‘₯π‘˜<1,1β‰€π‘˜β‰€π‘›) with boundary 𝑆,Ξ©=Ξ©βˆͺ𝑆. In [βˆ’1,1]Γ—Ξ©, the mixed boundary value problem for multidimensional mixed equation: βˆ’π‘’π‘‘π‘‘βˆ’π‘›ξ“π‘Ÿ=1ξ€·π‘Žπ‘Ÿ(π‘₯)𝑒π‘₯π‘Ÿξ€Έπ‘₯π‘Ÿπ‘’=𝑔(𝑑,π‘₯),0<𝑑<1,π‘₯∈Ω,𝑑+π‘›ξ“π‘Ÿ=1ξ€·π‘Žπ‘Ÿ(π‘₯)𝑒π‘₯π‘Ÿξ€Έπ‘₯π‘Ÿ=𝑓(𝑑,π‘₯),βˆ’1<𝑑<0,π‘₯∈Ω,𝑓(0,π‘₯)+𝑔(0,π‘₯)=0,𝑓(βˆ’1,π‘₯)+𝑔(1,π‘₯)=0,π‘₯∈Ω,𝑒(𝑑,π‘₯)=0,π‘₯βˆˆπ‘†,βˆ’1≀𝑑≀1;𝑒(1,π‘₯)=𝑒(βˆ’1,π‘₯)+πœ‡(π‘₯),π‘₯∈Ω,𝑒(0+,π‘₯)=𝑒(0βˆ’,π‘₯),𝑒𝑑(0+,π‘₯)=𝑒𝑑(0βˆ’,π‘₯),π‘₯∈Ω(3.1) is considered. Here, π‘Žπ‘Ÿ(π‘₯)(π‘₯∈Ω),πœ‡(π‘₯)(πœ‡(π‘₯)=0,π‘₯βˆˆπ‘†),𝑔(𝑑,π‘₯)(π‘‘βˆˆ(0,1),π‘₯∈Ω), and𝑓(𝑑,π‘₯)(π‘‘βˆˆ(βˆ’1,0),π‘₯∈Ω) are given smooth functions and π‘Žπ‘Ÿ(π‘₯)β‰₯π‘Ž>0.

The discretization of problem (3.1) is carried out in two steps. In the first step, the grid sets ξ‚Ξ©β„Ž=ξ€½π‘₯=π‘₯π‘š=ξ€·β„Ž1π‘š1,…,β„Žπ‘›π‘šπ‘›ξ€Έξ€·π‘š,π‘š=1,…,π‘šπ‘›ξ€Έ,0β‰€π‘šπ‘Ÿβ‰€π‘π‘Ÿ,β„Žπ‘Ÿπ‘π‘Ÿξ€Ύ,Ξ©=1,π‘Ÿ=1,…,π‘›β„Ž=ξ‚Ξ©β„Žβˆ©Ξ©,π‘†β„Ž=ξ‚Ξ©β„Žβˆ©π‘†(3.2)

are defined. To the differential operator 𝐴 generated by problem (3.1), the difference operator 𝐴π‘₯β„Ž is assigned by formula: 𝐴π‘₯β„Žπ‘’β„Ž=βˆ’π‘›ξ“π‘Ÿ=1ξ‚€π‘Žπ‘Ÿ(π‘₯)π‘’β„Žπ‘₯π‘Ÿξ‚π‘₯π‘Ÿ,π‘šπ‘Ÿ(3.3) acting in the space of grid functions π‘’β„Ž(π‘₯), satisfying the conditions π‘’β„Ž(π‘₯)=0 for all π‘₯βˆˆπ‘†β„Ž. With the help of 𝐴π‘₯β„Ž, we arrive at the nonlocal boundary-value problem βˆ’π‘‘2π‘’β„Ž(𝑑,π‘₯)𝑑𝑑2+𝐴π‘₯β„Žπ‘’β„Ž(𝑑,π‘₯)=π‘”β„Ž(𝑑,π‘₯),0<𝑑<1,π‘₯βˆˆΞ©β„Ž,π‘‘π‘’β„Ž(𝑑,π‘₯)π‘‘π‘‘βˆ’π΄π‘₯β„Žπ‘’β„Ž(𝑑,π‘₯)=π‘“β„Ž(𝑑,π‘₯),βˆ’1<𝑑<0,π‘₯βˆˆΞ©β„Ž,π‘’β„Ž(βˆ’1,π‘₯)=π‘’β„Ž(1,π‘₯)+πœ‡β„Ž(Ωπ‘₯),π‘₯βˆˆβ„Ž,π‘’β„Ž(0+,π‘₯)=π‘’β„Ž(0βˆ’,π‘₯),π‘‘π‘’β„Ž(0+,π‘₯)=π‘‘π‘‘π‘‘π‘’β„Ž(0βˆ’,π‘₯)Ω𝑑𝑑,π‘₯βˆˆβ„Ž,(3.4) for an infinite system of ordinary differential equations.

In the second step, problem (3.4) is replaced by difference scheme (1.2) (see [21]): βˆ’π‘’β„Žπ‘˜+1(π‘₯)βˆ’2π‘’β„Žπ‘˜(π‘₯)+π‘’β„Žπ‘˜βˆ’1(π‘₯)𝜏2+𝐴π‘₯β„Žπ‘’β„Žπ‘˜(π‘₯)=π‘”β„Žπ‘˜π‘”(π‘₯),β„Žπ‘˜(π‘₯)=π‘”β„Žξ€·π‘‘π‘˜ξ€Έ,π‘₯,π‘‘π‘˜=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,π‘πœ=1,π‘₯βˆˆΞ©β„Ž,π‘’β„Žπ‘˜(π‘₯)βˆ’π‘’β„Žπ‘˜βˆ’1(π‘₯)πœβˆ’π΄π‘₯β„Žπ‘’β„Žπ‘˜βˆ’1(π‘₯)=π‘“β„Žπ‘˜π‘“(π‘₯),β„Žπ‘˜(π‘₯)=π‘“β„Žξ€·π‘‘π‘˜ξ€Έ,π‘₯,π‘‘π‘˜βˆ’1=(π‘˜βˆ’1)𝜏,βˆ’π‘+1β‰€π‘˜β‰€βˆ’1,π‘₯βˆˆΞ©β„Ž,π‘’β„Žβˆ’π‘(π‘₯)=π‘’β„Žπ‘(π‘₯)+πœ‡β„Žξ‚Ξ©(π‘₯),π‘₯βˆˆβ„Ž,π‘’β„Ž1(π‘₯)βˆ’π‘’β„Ž0(π‘₯)=π‘’β„Ž0(π‘₯)βˆ’π‘’β„Žβˆ’1Ω(π‘₯),π‘₯βˆˆβ„Ž.(3.5) To formulate the result, we introduce the Hilbert spaces 𝐿2β„Ž=𝐿2(ξ‚Ξ©β„Ž),π‘Š12β„Ž=π‘Š12(ξ‚Ξ©β„Ž), and π‘Š22β„Ž=π‘Š22(ξ‚Ξ©β„Ž) of the grid functions πœ‘β„Ž(π‘₯)={πœ‘(β„Ž1π‘š1,…,β„Žπ‘›π‘šπ‘›)} defined on ξ‚Ξ©β„Ž, equipped with the norms: β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž=βŽ›βŽœβŽœβŽœβŽξ“π‘₯βˆˆξ‚Ξ©β„Ž||πœ‘β„Ž||(π‘₯)2β„Ž1β‹―β„Žπ‘›βŽžβŽŸβŽŸβŽŸβŽ 1/2,β€–β€–πœ‘β„Žβ€–β€–π‘Š12β„Ž=β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž+βŽ›βŽœβŽœβŽœβŽξ“π‘₯βˆˆξ‚Ξ©β„Žπ‘›ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯π‘Ÿ|||2β„Ž1β‹―β„Žπ‘›βŽžβŽŸβŽŸβŽŸβŽ 1/2,β€–β€–πœ‘β„Žβ€–β€–π‘Š22β„Ž=β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž+βŽ›βŽœβŽœβŽœβŽξ“π‘₯βˆˆξ‚Ξ©β„Žπ‘›ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯π‘Ÿ|||2β„Ž1β‹―β„Žπ‘›βŽžβŽŸβŽŸβŽŸβŽ 1/2+βŽ›βŽœβŽœβŽœβŽξ“π‘₯βˆˆξ‚Ξ©β„Žπ‘›ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯π‘Ÿπ‘₯π‘Ÿ,π‘šπ‘Ÿ|||2β„Ž1β‹―β„Žπ‘›βŽžβŽŸβŽŸβŽŸβŽ 1/2.(3.6)

Theorem 3.1. Let 𝜏 and |β„Ž|=β„Ž21+β‹…β‹…β‹…+β„Ž2𝑛 be sufficiently small numbers. Then, the solutions of difference scheme (3.5) satisfy the following coercivity stability estimate: β€–β€–ξ€½πœβˆ’2ξ€·π‘’β„Žπ‘˜+1βˆ’2π‘’β„Žπ‘˜+π‘’β„Žπ‘˜βˆ’1ξ€Έξ€Ύ1π‘βˆ’1‖‖𝐢𝛼([0,1]𝜏,𝐿2β„Ž)+β€–β€–ξ€½πœβˆ’1ξ€·π‘’β„Žπ‘˜βˆ’π‘’β„Žπ‘˜βˆ’1ξ€Έξ€Ύ0βˆ’π‘+1‖‖𝐢𝛼/2([βˆ’1,0]𝜏,𝐿2β„Ž)+β€–β€–ξ€½π‘’β„Žπ‘˜ξ€Ύπ‘βˆ’1βˆ’π‘β€–β€–πΆπ›Ό([βˆ’1,1]𝜏,π‘Š22β„Ž)ξ‚Έβ€–β€–πœ‡β‰€π‘€β„Žβ€–β€–π‘Š22β„Ž+1‖‖𝑓𝛼(1βˆ’π›Ό)β„Žπ‘˜ξ€Ύβˆ’1βˆ’π‘+1‖‖𝐢𝛼/2([βˆ’1,0]𝜏,𝐿2β„Ž)+β€–β€–ξ€½π‘”β„Žπ‘˜ξ€Ύ1π‘βˆ’1‖‖𝐢𝛼([0,1]𝜏,𝐿2β„Ž)ξ‚„ξ‚Ή,(3.7) where 𝑀 is not dependent on 𝜏,β„Ž,πœ‡β„Ž(π‘₯),π‘”β„Žπ‘˜(π‘₯),1β‰€π‘˜β‰€π‘βˆ’1, and π‘“β„Žπ‘˜,βˆ’π‘+1β‰€π‘˜β‰€0.

The proof of Theorem 3.1 is based on Theorem 2.3, the symmetry properties of the difference operator 𝐴π‘₯β„Ž defined by formula (3.3), and along with the following theorem on the coercivity inequality for the solution of elliptic difference equation in 𝐿2β„Ž.

Theorem 3.2. For the solution of elliptic difference problem: 𝐴π‘₯β„Žπ‘’β„Ž(π‘₯)=πœ”β„Ž(π‘₯),π‘₯βˆˆΞ©β„Ž,π‘’β„Ž(π‘₯)=0,π‘₯βˆˆπ‘†β„Ž,(3.8) the following coercivity inequality holds [27]: π‘›ξ“π‘Ÿ=1β€–β€–ξ€·π‘’β„Žξ€Έπ‘₯π‘Ÿπ‘₯π‘Ÿ,π‘šπ‘Ÿβ€–β€–πΏ2β„Žβ€–β€–πœ”β‰€π‘€β„Žβ€–β€–πΏ2β„Ž.(3.9) Here, 𝑀 depends neither on β„Ž nor π‘€β„Ž(π‘₯).

Acknowledgments

The author would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for his inspirational contributions and to the anonymous referees whose careful reading of the paper and valuable comments helped to improve it.

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