Abstract and Applied Analysis
Volume 2007 (2007), Article ID 61659, 16 pages
doi:10.1155/2007/61659
A note on the parabolic differential and difference equations
Allaberen Ashyralyev1
, Yasar Sozen1
and Pavel E. Sobolevskii3
1Department of Mathematics, Fatih University, Istanbul 34500, Turkey
3Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel
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.