(untitled) (European Journal of Pure and Applied Mathematics, 2026)
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.