A class of nonlinear delay evolution equations with nonlocal initial conditions

Abstract:

We establish a sufficient condition for the existence, uniqueness and global uniform asymptotic stability of a $C^0$-solution for the nonlinear delay differential evolution equation \begin{equation*}\left \{\begin {array}{ll} \displaystyle u’(t)\in Au(t)+f(t,u_t),&\quad t\in \mathbb {R}_+, \\[1mm] u(t)=g(u)(t),&\quad t\in [ -\tau ,0 ], \end{array}\right .\end{equation*} where $\tau >0$, $X$ is a real Banach space, $A$ is the infinitesimal generator of a nonlinear semigroup of contractions, $f:\mathbb {R}_+\times C([ -\tau ,0 ];\overline {D(A)})\to X$ is continuous and $g:C_b([ -\tau ,+\infty );\overline {D(A)})\to C([ -\tau ,0 ];\overline {D(A)})$ is nonexpansive.