A continuum of solutions in a Fréchet space of a nonlinear functional integral equation in N variables

In this paper, we investigate the set of solutions of a nonlinear functional integral equation in N variables in a Fréchet space. Applying a fixed point theorem of Krasnosel'skii type and a structure theorem of Krasnosel'skii and Perov, a sufficient condition is established such that the set of solutions is a continuum, that is, nonempty, compact and connected. Furthermore, based on Aronszajn type results and a theorem proved by Vidossich, we show that this solutions set is also a compact . This is also true with solutions set of a nonlinear Volterra–Hammerstein integral equation.