An approximation solvability method for nonlocal semilinear differential problems in Banach spaces

A new approximation solvability method is developed for the study of semilinear differential equations with nonlocal conditions without the compactness of the semigroup and of the nonlinearity. The method is based on the Yosida approximations of the generator of C0semigroup, the continuation principle, and the weak topology. It is shown how the abstract result can be applied to study the reaction-diffusion models.