Abstract and Applied Analysis 
Volume 2007 (2007), Article ID 61659, 16 pages
doi:10.1155/2007/61659
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

Recommended by Martin J. Bohner

Abstract

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(ℝ,E) with norm ‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder space Cα(E)=Cα(ℝ,E) with norm ‖ϕ‖Cα(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(ℝτ,E) spaces is proved. The well-posedness of this difference scheme in Cα(ℝτ,E) spaces is obtained.