On a nonlinear Volterra integral equation in a Banach space

AbstractThe equation u(t) + ∝0tk(t − s)g(s) ds ϵ f(t), t ⩾ 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g(t) and the function f. The case when k is realvalued, g(t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy