Convergence Results for Suzuki Generalized Nonexpansive Maps Using the HK-Iteration Scheme

Abstract

Over the years, enhancing classical iteration methods by blending them with Picard’s approach has led to significantly improved convergence outcomes, as seen in schemes like the Picard-Noor, Picard-S, Picard-Ishikawa, and Picard-Mann hybrids. Motivated by this trend, the present work introduces a novel three-step iterative scheme, termed the HK-Iteration. This method incorporates specific control sequences and builds on a structured composition of operator evaluations. We demonstrate the efficiency of the HK-Iteration through a comparative numerical example. Furthermore, we establish both strong and weak convergence results for Suzuki generalized nonexpansive mappings under the proposed iteration. As an application, we extend the HK-Iteration framework to approximate solutions of delay differential equations.