Common Fixed Points of Triplet Mappings in GM-Spaces with Applications to AI Convergence and Cryptographic Consensus (European Journal of Pure and Applied Mathematics, 2025)
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 determinewhether 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.