Lacunary statistical convergence in A-metric space via modulus function

Abstract

This paper systematically develops the theory of m-lacunary statistical convergence and strong m-lacunary summability of order ? within the framework of A-metric spaces, a higher-order generalization of ordinary metric spaces wherein distances are measured among n-tuples of points. By combining an unbounded modulus function m, a lacunary sequence ?=(tr), and a real order parameter ??(0,1], we introduce the sequence spaces AS?m,? and A??m,?, along with the related variants A??[?,m], A??,mm,?, and A??,m?. Precise inclusion relations among these spaces and their connections to classical lacunary statistical, m-statistical, and ordinary statistical convergence are established. The results highlight the role of conditions such as the growth rate of the modulus function, boundedness of the A-metric space, and lacunary refinement in governing convergence behavior. The results unify and extend several strands of summability theory to the very general setting of A-metric spaces, highlighting the intricate interplay between the modulus growth, the lacunary block structure, the order parameter, and the boundedness of the ambient space. As an application, the developed theory is applied to fixed point iterations in A-metric spaces, showing that Picard iteration sequences generated by contraction mappings satisfy suitable m-lacunary statistical convergence and summability conditions. The paper thereby provides a foundational hierarchy of convergence notions with potential applications in fixed point theory, stability analysis of iterative methods, and functional analysis on generalized metric structures. © The Author(s) under exclusive license to Università degli Studi di Ferrara 2026.