Calculus for interval-valued functions using generalized Hukuhara derivative and applications

• Published in: Fuzzy sets and systems 2013-05, Vol.219, p.49-67
• Main Authors: Chalco-Cano, Y., Rufián-Lizana, A., Román-Flores, H., Jiménez-Gamero, M.D.
• Format: Article
• Language: English
• Subjects: Differentiability; Fuzzy functions; Generalized Hukuhara difference; Integrability; Interval-valued functions
• ISSN: 0165-0114; 1872-6801
• DOI: 10.1016/j.fss.2012.12.004

This paper is devoted to studying differential calculus for interval-valued functions by using the generalized Hukuhara differentiability, which is the most general concept of differentiability for interval-valued functions. Conditions, examples and counterexamples for limit, continuity, integrability and differentiability are given. Special emphasis is set to the class F(t)=C·g(t), where C is an interval and g is a real function of a real variable. Here, the emphasis is placed on the fact that F and g do not necessarily share their properties, underlying the extra care that must be taken into account when dealing with interval-valued functions. Two applications of the obtained results are presented. The first one determines a Delta method for interval valued random elements. In the second application a new procedure to obtain solutions to an interval differential equation is introduced. Our results are relevant to fuzzy set theory because the usual fuzzy arithmetic, extension functions and (mathematical) analysis are done on α-cuts, which are intervals.