Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations (Q812703)

This paper continues the work of the first author and \textit{H. R. Henríquez} [J. Math. Anal. Appl., 221, 452--475 (1998; Zbl 0915.35110), J. Math. Anal. Appl. 221, No.2, 499--522 (1998; Zbl 0926.35151)] for partial neutral functional-differential equations with unbounded delay, where they considered the existence of solutions and periodic solutions of the given equation. Let \(X\) be a Banach space, \(x_t:(-\infty,0]\to X,x_t(\theta)=x(t+\theta)\), belongs to some abstract phase space \(\mathcal{B}\). The authors consider the existence of asymptotically almost-periodic and almost-periodic (mild) solutions for a class of neutral functional-differential equations with unbounded delay modelled in the form \[ \frac{d}{dt}(x(t)+f(t,x_t))=Ax(t)+g(t,x_t), \] where \(A\) is the infinitesimal generator of a uniformly exponentially stable analytic semigroup on \(X\).