Logarithmically Weighted Κ-Contractions in Fréchet Spaces: Existence and Uniqueness For Nonlinear Integral Equations with Fractional Dynamics

Abstract

This paper presents a novel theoretical framework to establish the existence and uniqueness of solutions for nonlinear functional integral equations. The analysis is conducted within the intricate topological structure of Fréchet spaces, which presents a significant generalization beyond the more common Banach space settings. The central contribution is the formulation of a logarithmically weighted -contraction condition, defined as . This specific parameter optimally incorporates the Lipschitz constants , the nonlinear function bounds , and an explicitly derived decay factor associated with the seminorm structure. The methodological paradigm navigates the core topological challenges inherent in non-normable spaces by constructing a convergent sequence of solutions on bounded intervals, and subsequently extending the global solution through a rigorous argument of consistency. The mathematical optimality of the contraction parameter  is analytically demonstrated by showing that the weight function, expressed as , attains its global maximum value of  at the specific point where . The theoretical construct is further validated through a fractional generalization utilizing Mittag-Leffler kernels, and a detailed case study is presented on a physical model of heat transfer that incorporates oscillatory and saturating memory effects. This comprehensive solution theory effectively bridges abstract principles of functional analysis with practical modeling applications for hereditary phenomena in infinite-dimensional systems.