Stable and unstable manifolds for nonlinear partial neutral functional differential equations

Abstract

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic equilibria for nonlinear partial neutral functional differential equations. We suppose that the linear part $A$ satisfies the Hille-Yosida condition on a Banach space and is not necessarily densely defined; the delayed part is assumed to be Lipschitz. We show the existence of stable and unstable manifolds near hyperbolic equilibria when the neutral operator is stable and the semigroup generated by the part of $A$ in $\overline{D(A)}$ is compact. Local stable and unstable manifolds are also obtained when the undelayed part is a C$^{1}$ function in a neighborhood of the equilibria.