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.

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.