Existence of the solution of a nonlinear integro-differential equation (Q1263788)

The author considers the scalar integral equation \(du/dt+a(t)u(t)+\int^{t}_{0}k(t,s)u(t-s)u(s)ds=f(t),\quad 0\leq t\leq T,\quad u(0)=c,\) and shows that the unique global solution can be obtained by an iterative sequence. The key assumption is that the kernel k is of type \(L^{\infty}\) on [0,T], i.e., that \(\sup_{0\leq t\leq T}\int^{t}_{0}| k(t,s)| ds<\infty\).