Common Fixed Points of Triplet Mappings in GM-Spaces with Applications to AI Convergence and Cryptographic Consensus

Abstract

This paper investigates the existence and uniqueness of common fixed points for three self-mappings in generalized metric spaces (GM-spaces), establishing new results under generalized contractive conditions. We develop a comprehensive theoretical framework where tripartite mappings satisfy inequalities involving combinations of distance-like terms formulated through minimum and maximum comparisons. The contractive conditions are governed by carefully chosen parameters that determine
whether the mappings admit a common fixed point or a unique common fixed point. To demonstrate the practical applicability of our theory, we present a concrete example using the interval [0, 1] equipped with the G-metric G(x, y, z) = max{|x − y|, |y − z|, |z − x|}, where piecewise-defined self-mappings are shown to satisfy all required conditions and converge to a unique common fixed point. Beyond theoretical advancements, our results offer significant applications in artificial intelligence, particularly in analyzing convergence of multi-layer neural architectures, and in cryptography for designing secure iterative protocols. The framework presented here not only generalizes existing fixed-point theorems but also provides verifiable computational methods for stability analysis in both mathematical and applied contexts.