A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space

The Bitsadze-Samarskii type nonlocal boundary value problem−u″(t)+Au(t)=f(t)(0⩽t⩽1),u(0)=φ,u(1)=u(λ)+ψ,0⩽λ<1, in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauder's estimates of solutions of the boundary value problems for elliptic equations are obtained.