Nonlinear abstract boundary-value problems in vector-valued function spaces and applications

The nonlocal boundary value problems for linear and nonlinear differential-operator equations of second order, with dependent coefficients, are studied. The principal part of the differential operators generated by linear problems are nonself-adjoint. Several conditions for the maximal regularity, RR-positivity and the fredholmness in Banach-valued LpLp-spaces are given. By using these results, the existence and uniqueness of the maximal regular solutions of nonlocal boundary value problems for nonlinear differential-operator equations are established. In applications, nonlocal boundary-value problems for nonlinear partial differential equations and their finite or infinite systems on cylindrical domains are studied.