Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

The nonlocal boundary value problem for the parabolic differential equation v' (t)+ A(t) v(t) = t(t) (0 <= t <= T), v(0) = v(lambda)+phi 0 < lambda <= T in an arbitrary Banach space.. with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces , 0 (t - tau)(gamma) of all E alpha-beta-valued continuous functions phi(t) on [0, T] satisfying a Holder condition with a weight (t + tau)(gamma). New Schauder type exact estimates in Holder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.