Almost periodic solutions of Volterra equations and attractivity
Consider the system of nonlinear integral equations \[ x(t) = a(t) + \int^t_\alpha D \bigl( t,s,x (s) \bigr) ds, \tag{1} \] where \(\alpha\) is either finite or \(- \infty\), and the related system of limiting equations \[ x(t) = p(t) + \int^t_{- \infty} P \bigl( t,s,x (s) \bigr) ds. \tag{2} \] The function \(a\) is asymptotically almost periodic with almost periodic part \(p\), and the two functions \(D\) and \(P\) are related to each other in a similar manner. The paper contains several results on the relation between the solutions of these two equations. Among others, it is shown that a unique bounded solution of (2) must be almost periodic and a unique bounded solution of (1) must be asymptotically almost periodic, and that the two solutions have the same almost periodic part. The existence, uniqueness, and attractivity of an (asymptotically) almost periodic solution is also studied.
scientific article; zbMATH DE number 902519