Asian Journal of Pure and Applied Mathematics

Fixed point iterative methods are fundamental computational tools in nonlinear analysis,yet classical schemes often suffer from slow convergence and sensitivity to parameter selection. This study addresses these limitations by developing a novel Adaptive Implicit Iteration Scheme (AIIS) for approximating fixed points of semigroups of Lipschitzian hemicontractive-type operators in Banach spaces. Our approach extends existing theory by first establishing weak and strong convergence results for a standard implicit scheme applied to the generalized classes of Lipschitzian hemicontractive and α-hemicontractive semigroups. The primary innovation, however, lies in the AIIS framework, which dynamically adjusts the iteration parameter αn through a feedback mechanism based on local operator behavior, specifically αn = ϕ(|xn−1−T(tn)xn−1|). We prove robust convergence theorems for this adaptive scheme under mild conditions and provide comprehensive numerical simulations demonstrating its superior performance. Results show that AIIS achieves significantly faster convergence rates and enhanced robustness compared to conventional methods with static parameters. This research represents a paradigm shift from static iterative procedures to intelligent, adaptive computational methods with substantial implications for solving nonlinear operator equations in optimization, differential equations, and applied mathematics.