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 ) = f ( t ) ( 0 ≤ t ≤ T ) , v ( 0 ) = v ( λ ) + φ , 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A ( t ) is investigated. The well-posedness of this problem is established in Banach spaces C 0 β , γ ( E α - β ) of all E α - β -valued continuous functions φ ( t ) on [ 0 , T ] satisfying a Hölder condition with a weight ( t + τ ) γ . New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.