Hukuhara differentiability of interval-valued functions and interval differential equations on time scales Vasile Lupulescu ⇑ Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania Government College University, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, Pakistan article info Article history: Received 10 March 2011 Received in revised form 14 February 2013 Accepted 5 June 2013 Available online 13 June 2013 Keywords: Time scales Interval-valued function Generalized Hukuhara differentiability Interval differential equation abstract Using the concept of the generalized Hukuhara difference, in this paper we introduce and study the differentiability and the integrability for the interval-valued functions on time scales. Some illustrative examples to interval differential equations on time scales are presented. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction There are several mathematical models to study the behavior of the real world systems such as: static or dynamic, linear or nonlinear, continuous or discrete, deterministic or probabilistic. In many cases the knowledge about the parameters of a real world system is imprecise or uncertain because, generally, we cannot observe or measure these parameters with accu- rate. In these situations, the parameters cannot be represented by real numbers. This shortcoming is overcome using fuzzy or interval models. Interval analysis is based on the representation of an uncertain variable as an interval of real numbers. Interval analysis has a long history, but a new era of this area started with Moore’s monograph [56]. With this monograph it began a new era of applications to error analysis for digital computers. There is a large literature on the subject [3,4,7,36,40,53,57–61,65]. The connection between Fuzzy Theory and Interval Mathematics leads to the interval-valued fuz- zy set theory, which has been studied from different viewpoints by some authors [19,22,24,55]. The notion of fuzzy set was introduced by Zadeh [69] as an extension of the classical notion of set. The theory of fuzzy sets has developed in parallel with the theory of set-valued functions. Thus using the concepts of integral and Hukuhara derivative for set-valued functions, introduced by Aumann [6] and Hukuhara [31] respectively, similar concepts for fuzzy functions were introduced and studied by several authors [23,33,34,63,64,66]. Hukuhara derivative was the starting point for the topic of set differential equations (see [39] for basic concepts and references) and later also for fuzzy differential equations. Fuzzy differ- ential equations were first studied by [66], and they are the natural generalizations of set differential equations and a natural way to model uncertainty of dynamical systems. The approach based on Hukuhara derivative has the disadvantage that any solution of a fuzzy differential equation has increasing length of its support. Consequently, this approach cannot really reflect any of the rich behavior of ordinary differential equations [20]. Bede and Gal [8] (see also [9,15,16,35,67,68]), solved this shortcoming by introducing the notion of strongly generalized differentiability of fuzzy-number-valued functions. Here the 0020-0255/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.06.004 ⇑ Address: Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania. Tel.: +40 253225881. E-mail address: vasile@utgjiu.ro Information Sciences 248 (2013) 50–67 Contents lists available at SciVerse ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins