Locally relatively continuous perturbations of analytic semigroups and their associated evolution equations

This paper is concerned with nonlinear perturbations of analytic semigroups in general Banach spaces. Such perturbations, denoted B, are assumed to be locally relatively continuous in the sense that they are continous with respect to the graph norm of a fractional power of the generator A of the analytic semigroup. Here, the concept of local is defined through a secondary bornology on the original Banach space which is specified in terms of multiple lower semicontinuous functionals. Our main result is the equivalence of existence of so-called mild solutions to a class of evolution equations subject to certain growth conditions, and a number of subtangential conditions on the semilinear operator A+B. These include explicit subtangential conditions, specified using the graph norm of fractional powers of A, and an implicit type condition. Such implications may be called generation theorems for the nonlinear semigroups providing mild solutions to the class of evolution equations. Moreover, under the further assumption that the lower semicontinuous functionals are convex, it is shown that the conditions mentioned above are in fact equivalent to the existence of resolvents of the semilinear operator A+B, and hence a characteriza-tion theorem is obtained for nonlinear semigroups associated with evolution equations governed by the class of operators under consideration.