A Localized Fixed Point Framework for Uncertain Dynamic Programming in Fuzzy S-Metric Spaces
DOI:
https://doi.org/10.64229/aj542a10Keywords:
Fuzzy S-metric space, Local fixed point theorem, α-contractive mapping, Admissible mapping, Value iteration, Dynamic programming, Convergence analysis, Uncertainty modelingAbstract
In this paper, a local fixed point theorem for -contractive mappings in fuzzy S-metric spaces is proved. The result is stated on a closed ball and assumes a localized contractive condition along with an admissibility condition at the beginning. These conditions ensure that the closed ball is mapped to itself by the given function and the iterative sequence related to the function converges to a unique fixed point in the same environment. The localized contractive condition allows the fixed point theorem to be applied in contexts where the contractive condition might not be valid, thus broadening the applications of fixed point theory in fuzzy S-metric spaces. To show the efficiency and applicability of the proposed outcomes of the theory, a real-life problem that occurs in the value iteration process in dynamic programming is provided in the following scenario. In this problem, the fuzzy S-metric theory is capable of incorporating the uncertainties and imprecise concepts of the reward function or transition mechanism, usually found in real-life problems of this category. The results attained within this study offer a robust basis for analysis on the convergence properties of computational methods, optimization processes, and iterative algorithms under uncertainty conditions. In addition, it further enhances the advancements of fixed points theory within fuzzy S-metric spaces and provides new avenues for applications within the areas of applied mathematics and related disciplines.
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