## Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations.(English)Zbl 1061.26024

Summary: The usual concept of differentiability of fuzzy-number-valued functions has the following shortcoming: if $$c$$ is a fuzzy number and $$g:[a,b]\to \mathbb R$$ is a usual real-valued function differentiable in $$x_{0}\in (a,b)$$ with $$g'(x_{0})\leq 0$$, then $$f(x)=c\odot g(x)$$ is not differentiable in $$x_{0}$$. In this paper we introduce and study generalized concepts of differentiability (of any order $$n \in \mathbb N)$$, which solves this shortcoming. A Newton-Leibniz-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 $$c\odot g(x)$$ are given.

### MSC:

 26E50 Fuzzy real analysis 34A99 General theory for ordinary differential equations 35A99 General topics in partial differential equations
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