Network-Type and Measure-Functor Properties of the Space of Solutions of Nonlinear Volterra–Fredholm Integro-Differential Equations.

Abstract

We study topological and network-type properties of the solution space of a nonlinear Volterra–Fredholm integro-differential equation on a compact interval. The equation contains both a Volterra memory term and a Fredholm nonlocal term and is considered under continuity assumptions on all nonlinearities and kernels. The space of classical solutions is endowed with the topology inherited from the Banach space of continuous vector-valued functions with the uniform norm.

The first main result shows that the solution space is closed in the ambient Banach space. Consequently, it is a Polish space. A Polish space is a separable, completely metrizable topological space. We then prove a stronger structural fact: the solution space is σ-compact. Indeed, its intersection with every uniform norm ball is compact; this follows from uniform bounds for the integral terms, a uniform derivative estimate, and the Arzelà–Ascoli theorem. These compactness properties remain valid in the genuinely nonlocal Volterra–Fredholm setting despite the presence of memory and global feedback.

As consequences of the Polish and second countable structure, the solution space has a countable network, a countable cs-network, a countable cs*-network, a countable cn-network, and a countable ck-network. We further investigate measure-functor constructions. The space of all Borel probability measures on the solution space, equipped with the weak topology, is Polish and hence possesses all of the above countable network-type properties. The same is true for its finite-support subspaces. For each compact bounded stratum of the solution space, the spaces of probability measures and idempotent probability measures are compact metrizable spaces; therefore their network, cs-network, cs*-network, cn-network, and ck-network are countable. Functorial consequences under continuous transformations of solution spaces are also recorded.

The results extend the topological program previously developed for spaces of solutions of ordinary differential equations to a genuinely nonlocal Volterra–Fredholm setting and reveal additional Polish and compactness features produced by the integral structure.