Uniqueness and global convergence of successive approximations for solutions of functional integral equations with infinite delay (Q1082569)

Consider the abstract integral equation \(x(t)=Lx(t)\), where \(Lx(t)=f(t)\), \(t\leq a\); \(Lx(t)=f(t)+\int^{t}_{a}g(t,s,x_ s)ds\), \(t>a\); \(x_ s(\mu)=x(s+\mu)\), \(\mu\leq 0\). Uniqueness of the solution of this equation is proved as well as global convergence of the successive approximations \(x_ n(t)=Lx_{n-1}(t)\). Thus previous results of \textit{H. Chen} [ibid. 80, 19-30 (1981; Zbl 0506.45014)] are extended and generalized.