Almost Periodic Solutions for Nonlinear Delay Evolutions with Nonlocal Initial Conditions

Ioan Vrabie*

We consider a nonlinear delay differential evolution equation of the form \begin{equation*}\left\{\begin{array}{ll}\label{eqa.1} \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 $A:D(A)\subseteq X\leadsto X$ is an $m$-dissipative operator in the Banach space $X$, $\tau\geq 0$, $f:\mathbb{R}_+\times C([\,-\tau,0\,];\overline{D(A)})\to X$ is jointly continuous and Lipschitz with respect to the second argument and $g:C_b([\,-\tau,+\infty);\overline{D(A)})\to C([\,-\tau,0\,];\overline{D(A)})$ is nonexpansive. We prove that if $(I-A)^{-1}$ is compact, the nonlinear semigroup generated by $A$ decays exponentially and the Lipschitz constant of $f$ is sufficiently small, then the unique $C^0$-solution of the problem above is almost periodic.

Mathematics Subject Classification: 34K14 47H06 47J35

Keywords: delay equations; nonlocal initial conditions; almost periodic solutions

Minisymposion: Analysis and Control of Evolution Equations and Inclusions