In general, we have constructed the operators ideal generated by extended s-fuzzy numbers and a certain space of sequences of fuzzy numbers. An investigation into the conditions sufficient for Nakano sequence space of fuzzy numbers furnished with the definite function to create pre-quasi Banach and closed is carried out. The (R) and the normal structural properties of this space are shown. Fixed points for Kannan contraction and non-expansive mapping have been introduced. Lastly, we explore whether the Kannan contraction mapping has a fixed point in its associated pre-quasi operator ideal. The existence of solutions to non-linear difference equations is illustrated with a few real-world examples and applications.

Since proving many fixed point theorems in a given space requires either growing the space itself or growing the selfmapping that works on it, both of these options are good. The operators' ideal generated by a weighted binomial matrix in the Nakano sequence space of extended s-fuzzy functions is constructed. Some structures for it based on geometry and topology are presented. It has been proven that the Kannan contraction operator has a unique fixed point in this class. Lastly, sufficient conditions such that a fuzzy non-linear matrix system of Kannan-type has a unique solution in this ideal class are investigated and a numerical example to explain our results are given.