Fixed-point topology meets fractal memory: a Kutumba-stabilized framework for nonlocal fractal–fractional dynamics

This paper presents an innovative synthesis of generalised fixed-point theory, advanced topological degree methodologies, and high-order computational frameworks for memory-driven dynamical systems. We expand contraction principles in complete metric and ANR spaces, enhancing existence results through the Leray-Schauder degree in the context of non-compact perturbations. We develop a high-order numerical framework for fractal–fractional differential equations employing the Taylor Operational Matrix Method (TOMM) and an adaptive Adams–Bashforth–Moulton (ABM) scheme. This framework leverages global basis approximations to achieve high precision and has strong stability guarantees thanks to the Ulam–Hyers–Rassias criteria and Lyapunov-Razumikhin functionals. This framework has strong stability guarantees thanks to the Ulam–Hyers–Rassias criteria and Lyapunov-Razumikhin functionals. We incorporate sociomathematical structures, such as Kutumba-inspired familial support, into epidemic models, illustrating how topological and fractional tools collectively encapsulate memory, heterogeneity, and resilience. Applications encompass fractional epidemiology, biomedical hysteresis, and cyber-virus dynamics, demonstrating interdisciplinary effectiveness. The work connects abstract analysis with real-world complexity, giving a single method for nonlocal, socially embedded systems.