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

Bede, Barnabás; Gal, Sorin G.

Fuzzy Sets Syst. 151, No. 3, 581-599 (2005).

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

Keywords:

fuzzy-number-valued functions; generalized differentiability; ordinary fuzzy differential equations; fuzzy partial differential equations

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