Nonlinear Volterra integral equations and the Schröder functional equation (Q608395)

The author shows an interesting connection between a special class of Volterra integral equations with convolution kernels \[ u(t)=\int\limits_{0}^{t}k(t-s)g(u(s)ds, \quad g(0)=0, \quad t\geq 0, \] and the famous Schröder equation \[ F(h(x))=cF(x),\quad x\in I. \] Here \(h:[0,\infty)\to [0,\infty)\) is a given function; \(c>0\), \(c\neq 1\); \(F:I\to \mathbb{R}\) is an unknown function; \(I\) is a given interval such that \(h(I)\subset I\). The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented.