Unique solvability of nonlinear Volterra equations in weighted spaces

We investigate integral equations of the form x(t) = g(t) + ∫_-∞^t F(t, s, x(s)) ds. (^*) In general, this equation is history-dependent, so one needs to give an initial condition on (-∞, 0] in order to obtain a unique solution. By introducing a weight function on R, we can single out a class of admissible solutions, and give conditions for the unique solvability of (^*) in this restricted class. We also study some Fredholm equations on these weighted spaces. In addition, we also treat a class of equations of the first kind for which similar conclusions can be drawn.