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In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

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.