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(λ) + φ, 0 < λ ≤ 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 β,γ 0 (E) of all continuous functions E-valued functions φ(t) on [0, T] satisfying a H¨older condition with a weight (t+τ ) γ . New exact estimates in Holder norms for the solution of three nonlocal-boundary value problems for parabolic equations are obtained.