On the Solvability of Hybrid Ψ-Caputo Langevin Equations Featuring Memory Effects and Non-Local Conditions

Abstract

This research investigates the existence and uniqueness of solutions for a novel class of hybrid
integrodifferential Langevin equations governed by the Ψ-Caputo fractional derivative. The proposed model
distinguishes itself by integrating a multiplicative hybrid non-linear structure with a constant time-delay and
a Volterra-type integral term. To account for the system’s hereditary characteristics, a continuous history
condition is employed, characterizing the state trajectory prior to the initial process commencement. By
applying Dhage’s fixed-point theorem in Banach algebras and the Banach contraction principle, we establish
the fundamental criteria required to guarantee the existence of a unique state evolution. This mathematical
framework is particularly effective for modeling complex dynamics in viscoelasticity and biological systems,
where processes are simultaneously influenced by discrete delays and cumulative memory accumulation. A
numerical illustration, focused on industrial thermal dynamics, is included to demonstrate the consistency and
applicability of the theoretical results.