Ishikawa iteration for double sequences via predecessor paths and RH-regular matrix transforms

Abstract

This paper extends the Ishikawa fixed-point iteration scheme to double sequences through a diagonal processing framework with unique predecessor paths. We establish monotonicity along these paths and impose parameter conditions on boundary sequences to prove P-convergence of the iterative double sequences to fixed points of compact nonexpansive mappings in Hilbert space. Our results show that convergence extends from boundaries to interior points via diagonal ordering that resolves the two-step dependency structure. We prove that four-dimensional RH-regular matrix transforms preserve both P-convergence and asymptotic regularity, providing a unified treatment that integrates predecessor paths with RH-regular matrix transforms for extending Ishikawa iteration to double sequences.