Nonlinear evolution equations and product stable operators on Banach spaces

Abstract

The method of product integration is used to obtain solutions to the time dependent Banach space differential equation $u’(t) = A(t)(u(t)),t \geqq 0$, where $A$ is a function from $[0,\infty )$ to the set of nonlinear operators from the Banach space $X$ to itself and $u$ is a function from $[0,\infty )$ to $X$. The main requirements placed on $A$ are that $A$ is $m$-dissipative and product stable on its domain. Applications are given to a linear partial differential equation, to nonlinear dissipative operators in Hilbert space, and to continuous, $m$-dissipative, everywhere defined operators in Banach spaces.