Random Fixed Point Theorems for Relaxed Asymptotic Contractions in Random Normed Modules

Abstract

We introduce the notion of a random relaxed asymptotic contraction in the setting of random normed modules. The contraction condition employs two quasi-metrics that are built directly from the random operator: a lower quasi-metric which adaptively switches between a four-point minimum and the ordinary one-step distance, and an upper quasi-metric which takes the maximum of four fundamental distances. The bounds are allowed to depend on the iteration index and are required to converge locally uniformly almost surely to a Boyd--Wong function. Using the fibre decomposition method based on \(\sigma\)-stability and the local property, we show that any such mapping defined on an essentially bounded, \(\sigma\)-stable and \(L^0\)-closed set admits a unique random fixed point, and all iterates converge in the \((\epsilon,\lambda)\)-topology. Our result strictly generalizes the random analogue of Kirk's asymptotic contraction theorem and unifies several deterministic and random fixed point theorems under a single flexible framework.