A Class of Nonlinear Evolution Equations in a Banach Space

where the unknown function/is from a real number interval into a Banach space X. For suitable real numbers t and vectors x in X, Ait, x) is the infinitesimal generator of a holomorphic semigroup of linear contraction operators in X, and certain regularity requirements are placed on the function (/, x) -> Ait, x). After proving a local existence, uniqueness, and stability theorem for (*), we consider the case Ait, x) = Hix) and obtain conditions under which there is a strongly continuous semigroup of nonlinear nonexpansive transformations whose infinitesimal generator is an extension of the transformation Qx = H(x)x. We state our main results in §1 and prove them in §2. In §3, we prove some theorems about linear semigroups in a function space which yield examples of our main results and are of some interest in themselves.