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

机译：使用广义Hukuhara导数的区间值函数微积分及其应用

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摘要

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.
机译：本文致力于使用广义Hukuhara可微性研究间隔值函数的微积分，这是区间值函数可微性的最一般概念。给出了极限，连续性，可积性和可微性的条件，例子和反例。特别强调设置为类F（t）= C·g（t），其中C是区间，g是实变量的实函数。这里，重点放在F和g不一定共享其属性这一事实上，这是处理间隔值函数时必须考虑的额外注意事项的基础。给出了所得结果的两种应用。第一个确定间隔值随机元素的Delta方法。在第二个应用程序中，引入了一种新的程序来获取区间微分方程的解。我们的结果与模糊集理论有关，因为通常的模糊算术，扩展函数和（数学）分析都是在作为区间的α割上进行的。

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