The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(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 spaceCα(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 inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.
A Note on the Parabolic Differential and Difference Equations