Coercive solvability of parabolic differential equations with dependent operators

Abstract

In the present paper the nonlocal-boundary value problem for the differential equation of parabolic type v ? (t) + A(t)v(t) = f(t) (0 ? t ? T), v(0) = v(?) + ?, 0 < ? ? T in an arbitrary Banach space with the linear positive operators A(t) is considered. The well-posedness of this problem is established in Banach spaces C ?,? 0 (E) of all continuous functions E-valued functions ?(t) on [0, T] satisfying a H¨older condition with a weight (t+? ) ? . New exact estimates in Holder norms for the solution of three nonlocal-boundary value problems for parabolic equations are obtained.