Fibonacci-Ishikawa iterative method in modular spaces for asymptotically non-expansive monotonic mathematical operators

In the context of modular function spaces, we propose and investigate the Fibonacci-Ishikawa iteration method applied to non-expansive, asymptotically monotonic mathematical operators. We establish both ω-convergence and ω-almost everywhere convergence of the Fibonacci-Ishikawa sequence, thereby extending several existing results in the literature. Furthermore, our findings highlight the convergence behavior of the sequence toward its ω-almost everywhere limit, particularly as it relates to the minimizing sequence of a given function. Also, a comparative analysis with classical Mann and Ishikawa iteration methods reveals that the Fibonacci-Ishikawa iteration exhibits superior convergence properties, rapidly approaching the fixed point in significantly fewer iterations. Finally, we discuss the possible applicability of practical relevance.