Bounded solutions and periodic solutions of integral equations

This paper considers the functional equation X(t) = F(t, Xt), X is an element of R-n It defines uniform boundedness and ultimate uniform boundedness of solution s and investigates the existence of periodic solutions for the functional e quation using Horn's asymptotic fixed point theorem. As an application, thi s paper also considers the integral equation X(t) = a(t) + integral(t-alpha)(t) G(t, s, X(s))ds. X is an element of R-n. Using Liapunov second method, this paper offers the sufficient conditions f or uniform boundedness and ultimate uniform boundedness of solutions and th e existence of periodic solutions for the integral equation. AMS (MOS) subj ect classification: 39B99, 45M10, 45M15.