Calculus of linear fuzzy-number-valued functions using the generalized derivative and the Riemann integral of fuzzy n-cell-number-valued functions

Abstract

We study the differentiability, integral, and calculus of linear fuzzy-number-valued functions. Special emphasis is placed on the linear fuzzy-number-valued function F˜(t)=u˜1f1(t)+u˜2f2(t)+⋯+u˜mfm(t), where u˜1,u˜2,…,u˜m denote fuzzy n-cell numbers and f1,f2,…,fm represent real functions of a real variable. The concepts of the limit and continuity of fuzzy n-cell-number-valued functions are defined, which are the basis for studying the calculus of linear fuzzy-number-valued functions. Using the fuzzy generalized difference introduced by Gomes and Barros (2015) [8], we define a generalized difference for fuzzy n-cell numbers. Then a generalized differentiability of fuzzy n-cell-number-valued functions is proposed, and a sufficient condition for generalized differentiability of linear fuzzy-number-valued functions is given by means of real function theory. Furthermore, a Riemann integral of fuzzy n-cell-number-valued functions is introduced, and some properties of the integral are discussed. Finally, the relationship between generalized differentiability and the Riemann integral of the linear fuzzy-number-valued function F˜(t)=u˜1f1(t)+u˜2f2(t)+⋯+u˜mfm(t) is derived.