Fixed point theorems in fuzzy modular metric spaces for non-expansive mappings with applications to neural operator stability

It is also the aim of the paper to introduce new fixed-point results in the context of fuzzy modular metric spaces in connection to non-expansive maps and how they can be applied to the stability of neural operators.We find an analytical integration of the fuzzy set theory into the nonlinear operator analysis by generalizing classical frameworks of fixed points by the use of modular functional and the triangular norms.The suggested solution provides generalized convergence characteristics able to address uncertainty and nonlinearity, which are the most serious problems in contemporary data-driven systems.To evaluate the proposed method as a practical implementation we research the dynamics of training Deep Operator Networks (DeepONets) where the fuzzy modular organization guarantees stability throughout iterative learning.Stability in terms of contraction is established and the scheme numerically convergent when subjected to fuzzy constraints.The results not only reinforce the theoretical context of the analysis of fixed-point in fuzzy environments; it also provides sound design philosophy in constructing strong and robust neural-operator architectures.