Some connections between the generalized Hukuhara derivative and the fuzzy derivative based on strong linear independence

Abstract

This article investigates some connections between the notions of the generalized Hukuhara derivative and the Ψ-derivative of fuzzy number-valued functions. The concept of Ψ-differentiability is defined on a fuzzy number-valued function φ in the form of φ(x)=ϱ1(x)A1+…+ϱn(x)An, where ϱ1,…,ϱn are n real-valued functions defined on an interval [a,b] and {A1,…,An} is a strongly linearly independent subset of fuzzy numbers. The Ψ-derivative of φ at some x is φ′(x)=ϱ1′(x)A1+…+ϱn′(x)An whenever the derivatives ϱ1′(x),…,ϱn′(x) exist. This article provides conditions for these two notions of derivatives of such functions to coincide. Moreover, under some weak conditions, we show that an arbitrary continuously gH-differentiable function defined on an interval [a,b] and its gH-derivative can be uniformly approximated as closely as desired by a Ψ-differentiable function and its Ψ-derivative, respectively. Finally, we apply these results to obtain numerical and analytical solutions of a simple fuzzy decay model described by a fuzzy initial value problem under gH-derivative.