Boundedness and periodicity of solutions of neutral functional differential equations with infinite delay (Q5905419)

The author considers the neutral functional differential equation (1) \((d/dt)D_{x_ t}=f(t,x_ t)\) and the neutral Volterra integro- differential equation (2) \((d/dt)(x(t)-\int^ t_{-\infty}C(t- s)x(s)ds)=A(t)x(t)+\int^ t_{-\infty} E(t,s)x(s)ds+e(t)\), where \(x\in\mathbb{R}^ n\), \(f: R\times B\to\mathbb{R}^ n\), \(B\) is a given phase space, \(D: B\to\mathbb{R}^ n\) is a linear and continuous function, \(A\), \(C\), \(E\) are continuous functions. He presents without proof sufficient conditions for the solutions of (1) to be uniformly bounded or \(B\)- uniformly ultimately bounded and sufficient conditions for the solutions of (2) to be \(C_ h\)-uniformly bounded and \(C_ h\)-uniformly ultimately bounded. Sufficient conditions for the existence of a \(T\)-periodic solution of (1) and (2) are also introduced.