λ-Statistical Convergence and Attractors in Hyperspace Dynamics

Multiplicative metric spaces provide a natural framework for analyzing problems involving exponential growth or ratio-based phenomena unlike the classical additive settings. In this paper, we introduce the concept of multiplicative λ-statistical convergence by using generalized de la Valle-Poussin means to accommodate sequences with infrequent yet substantial deviations. We utilize the logarithmic relationship between multiplicative and classical metrics to bridge the gap between these two frameworks which facilitates the transfer of analytical properties without relying solely on isomorphism. We characterize multiplicative λ-statistically Cauchy sequences and investigate their connection to boundedness and completeness within this topology. Furthermore, we provide a detailed analysis of statistical limit and cluster points to demonstrate that the set of cluster points forms a closed and compact set under specific conditions. These results extend classical summability theory and offer new methodologies for analyzing systems governed by multiplicative calculus.