A novel difference and derivative of linearly correlated fuzzy number-valued functions

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.

Newton’s divided gH-difference interpolation formula with unequal spacing for interval-valued function using Hukuhara difference

Interval interpolation formulae play a significant role to find the value of an unknown function at some points under interval uncertainty. The objective of this paper is to establish Newton’s divided interpolation formula for interval-valued functions using generalized Hukuhara difference of intervals. For this purpose, arithmetic of intervals, Hukuhara difference and its some properties and concept of interval-valued function have been discussed briefly. Using Hukuhara difference of intervals, the definition of Newton’s divided gH-difference for interval-valued function has been introduced. Then Newton’s divided gH-differences interpolation formula has been derived. Finally, with the help of some numerical examples, the proposed interpolation formula has been illustrated.

An Extension of the Carathéodory Differentiability to Set-Valued Maps

This paper uses the generalization of the Hukuhara difference for compact convex set to extend the classical notions of Carathéodory differentiability to multifunctions (set-valued maps). Using the Hukuhara difference and affine multifunctions as a local approximation, we introduce the notion of CH-differentiability for multifunctions. Finally, we tackle the study of the relation among the Fréchet differentiability, Hukuhara differentiability, and CH-differentiability.

Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales

This paper mainly deals with introducing and studying the properties of generalized nabla differentiability for fuzzy functions on time scales via Hukuhara difference. Further, we obtain embedding results on E n for generalized nabla differentiable fuzzy functions. Finally, we prove a fundamental theorem of a nabla integral calculus for fuzzy functions on time scales under generalized nabla differentiability. The obtained results are illustrated with suitable examples.

A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given.