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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 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 Hölder Spaces

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

Received 30 March 2012; Accepted 17 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 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 𝑑 2 𝑢 ( 𝑡 ) / 𝑑 𝑡 2 + s i g n ( 𝑡 ) 𝐴 𝑢 ( 𝑡 ) = 𝑔 ( 𝑡 ) , ( 0 𝑡 1 ) , 𝑑 𝑢 ( 𝑡 ) / 𝑑 𝑡 + s i g n ( 𝑡 ) 𝐴 𝑢 ( 𝑡 ) = 𝑓 ( 𝑡 ) , ( 1 𝑡 0 ) , 𝑢 ( 0 + ) = 𝑢 ( 0 ) , 𝑢 ( 0 + ) = 𝑢 ( 0 ) , a n d 𝑢 ( 1 ) = 𝑢 ( 1 ) + 𝜇 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 Hölder spaces without a weight is established. Moreover, as applications, coercivity estimates in Hölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.