Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations

The usual concept of differentiability of fuzzy-number-valued functions, has the following shortcoming: if c is a fuzzy number and g:[a,b]→R is an usual real-valued function differentiable on x0∈(a,b) with g′(x0)⩽0, then f(x)=c⊙g(x) is not differentiable on x0. In this paper we introduce and study generalized concepts of differentiability (of any order n∈N), which solves this shortcoming. Newton-Leibnitz-type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied. Also, some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c⊙g(x), are given.