Perron type theorems for functional differential equations with infinite delay in a Banach space (Q2378811)

The authors prove some results on the asymptotic behavior of solutions for functional differential equations with infinite delay of the form \[ x'(t)=Ax(t)+Lx_t+f(t,x_t). \] Here, \(A\) is the infinitesimal generator of a compact \(C_0\)-semigroup in a Banach space \(X\), \(\gamma\in \mathbb{R}\), \(C_{\gamma}:=\{\Phi\in C(\mathbb{R}^-;X)\mid \lim_{\theta\to -\infty}|\Phi(\theta)|_Xe^{\gamma\theta}=0\}\), \(L:C_{\gamma}\to X\) is a bounded linear operator and \(f:\mathbb{R}^+\times C_{\gamma}\to X\) is a nonlinear mapping.