Periodic solutions of nonlinear integral equations (Q1106423)

The author considers the existence of periodic solutions of the system \(x(t)=f(t)+\int^{t}_{-\infty}q(t,s,x(s))ds,\) where \(f(t+T)=f(t)\) and \(q(t+T,s+T,x)=q(t,s,x)\). Under the assumptions that the solutions are bounded in a certain sense involving weight functions and depend continuously on initial functions, the existence of a periodic solution is established with the aid of Horn's fixed point theorem. Some conditions are given that imply the assumptions on boundedness and continuous dependence.