Boundedness and periodicity of solutions of neutral functional differential equations with infinite delay (Q5906626)

In this extensive paper, the author sets up systematically the theory of boundedness and periodicity of the solutions for neutral differential equations with infinite delay, (1) \({d \over dt} Dx_ t = f(t,x_ t)\), and constructs functional \(V(t, \varphi)\) and adds concrete conditions to ensure that the solutions of neutral Volterra integro-differential equations \[ {d \over dt} (x(t) - \int^ t_{- \infty} C(t - s) x(s)ds) = A(t)x(t) + \int^ t_{-\infty} E(t,s) x(s)ds + e(t) \tag{2} \] are uniformly bounded and uniformly ultimately bounded and to ensure the existence of periodic solutions. In (1) and (2) we have \(x \in R^ n\), \(f : R \times B \to R^ n\) is a continuous function, \(B\) is a given phase space, and \(C,A\) and \(E\) are continuous functions. The space \(B\) is the space \(C((- \infty,0] \to R^ n)\), and \(B\) is a Banach space. The author assumes three axioms on the space \(B\). Also, the author considers the operator \(D\) defined by \(Dx_ t = x(t) - \int^ 0_{- \infty} C(- s) x_ t(s) ds\), and proves that the operator \(D\) is uniformly stable and uniformly asymptotically stable.