Qualitative analysis of Volterra-type integral equations with an infinite delay model in spaces of locally bounded κ-variation functions

Integral equations with memory effects and infinite delay arise naturally in applications such as ecology, neuroscience, and viscoelasticity, among others. In this work, we study three classes of equations in the space of functions of locally bounded [Formula: see text]-variation, which accommodates functions with mild singularities whose oscillatory behaviour is controlled in a weaker sense than that provided by classical bounded variation. We analyze a nonlinear Volterra integral equation, a generalization of the Volterra-Hammerstein type equation, and an equation with infinite delay associated with the dynamics of an isolated species. For the first two equations, we establish conditions ensuring existence, uniqueness, and prolongation of solutions, while for the infinite-delay model we additionally derive explicit bounds for the solutions.