Integral solutions of locally Lipschitz continuous functional-differential equations

Abstract

A local existence and uniqueness result for the functional differential equation in a Banach space $X$ $$ \begin{equation} x'(t)=f(t)x(t)+g(t)x_t, \quad x_0=\phi, \ x(0)=h, \ \{\phi,h\}\in L^1(-R,0;X)\times X \tag{FDE} \end{equation} $$ is obtained, for the case where the operators $f(t)$ satisfy only a local dissipativity condition and the operators $g(t)$ are only locally Lipschitz continuous. This is done by relating (FDE) to the evolution equation in ${L^1(-R,0;X)\times X}$ $$ \begin{equation} u'(t)=A(t)u(t),\quad u(0)=\{\phi , h\}, \tag{E} \end{equation} $$ where $$ \begin{align} &D(A(t)) = \{\{\phi,h\}\in L^1(-R,0;X)\times X ; \ \phi\in W^{1,1}(-R,0;X), h\in D(f(t)),\phi (0)=h\} \\ &(t)\{\phi ,h\} =\{\phi ',\, f(t)h+g(t)\phi\}. \end{align} $$ It is shown that if $u(t)$ is the limit solution of (E), then $u(t)=\{x_t,x(t)\}$, where $x(t)$ is the integral solution