Abstract and Applied Analysis
Volume 2007 (2007), Article ID 61659, 16 pages
Research Article

A Note on the Parabolic Differential and Difference Equations

Allaberen Ashyralyev,1 Yasar Sozen,1 and Pavel E. Sobolevskii2,3

1Department of Mathematics, Fatih University, Istanbul 34500, Turkey
2Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel
3Universidade Federal do Ceará, Ceará 60020, Brazil

Received 13 August 2006; Revised 12 December 2006; Accepted 6 March 2007

Academic Editor: Martin J. Bohner

Copyright © 2007 Allaberen Ashyralyev et al. 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.


The differential equation u'(t)+Au(t)=f(t)(<t<) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach space C(E)=C(,E) with norm ϕC(E)=sup<t<ϕ(t)E. In the present paper, the well-posedness of this equation in the Hölder space Cα(E)=Cα(,E) with norm ϕCα(E)=sup<t<ϕ(t)E+sup<t<t+s<(‖ϕ(t+s)ϕ(t)E/sα), 0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in C(τ,E) spaces is proved. The well-posedness of this difference scheme in Cα(τ,E) spaces is obtained.