Generalized Hukuhara Differentiability of Interval-Valued Functions and Interval Differential Equations

Interval Analysis was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena: Interval Analysis, Reliable Computing, Validated Numerics, Interval problems with Differential Equations are discussed in several monographs and research papers. In classical Real Analysis, maybe one of the most important concepts is that of the derivative of a real-valued function. Correspondingly, in Interval Analysis or in the Theory of Differential Inclusions, we would expect to have a notion of the derivative of an interval-valued or set-valued function. Instead, the classical derivatives are used in both research directions which we have mentioned above. The reason for this is that a derivative concept which is both theoretically well founded and it is also applicable to concrete situations is still missing, despite the almost half a century of (otherwise very important) development of these domains. Hukuhara derivative of a set-valued mapping was first introduced by Hukuhara and it has been studied in several works. The paper of Hukuhara was the starting point for the topic of Set Differential Equations and later also for Fuzzy Differential Equations. Recently, several works have brought back into the attention of the nonlinear analysis community, the topics of set differential equations and the Hukuhara derivative. Also, as a very important generalization and development related to the subject of the present paper is in the field of fuzzy sets, i.e., fuzzy calculus and fuzzy differential equations. Hukuhara's differentiability concept has an important drawback, that is the paradoxical behavior of the solutions of a set or a fuzzy differential equation, i.e., "irreversibility under uncertainty". The paper shows that the generalization of the concept of Hukuhara differentiability can be of a great help in the study of interval differential equations (IDEs). The idea of the presented approach and differentiability concept comes from a generalization of the Hukuhara difference for compact convex sets (gH-difference) and the strongly and weakly generalized (Hukuhara) differentiability concepts. Combining these notions we obtain very simple formulations of the concepts and results with weakly generalized Hukuhara derivative (gH-derivative) by the help of the concept of gH-difference. The presented derivative concept is slightly more general than the notion of strongly generalized (Hukuhara) differentiability for the case of interval valued functions, it is actually equivalent with the concept of weakly generalized (Hukuhara) differentiability. In this paper, we prove several properties of the derivative concepts considered, we obtain examples of large classes of gH-differentiable functions and a characterization of gH-differentiable functions in terms of the differentiability of the real-valued functions giving the endpoint of the interval. Finally, we obtain existence of solutions for IDEs and characterization theorems for interval differential equations by ODEs.