This paper is devoted to make a framework for studying a class of uncertain differential equations called Z-differential equations. In order to achieve the purpose, we first introduce four basic operations on $Z^{+}$-numbers based on semigranular function. Then, the limit and continuity concepts of a Z-number-valued function are given, under a definition of a metric on the space of $Z^{+}$-numbers. Moreover, the concepts of Z-differentiability, Z-integral, and Z-Laplace transform of a Z-number-valued function are introduced. In addition, by giving some theories proved in this paper, a basis for calculus—Z-calculus—is established. We further give theories based on which existence and uniqueness of Z-differential equations are investigated. A conceptual unity between Z-differential equations and $Z^{+}$-numbers is also shown. The conceptual unity demonstrates that a Z-differential equation may be expressed as a bimodal differential equation combining a fuzzy differential equation (FDE) and a random differential equation. Moreover, the concept of a bimodal cut called $ (s, \mu)$-cut is introduced and its relation to other new concepts such as acceptable time and acceptable information area is explained. Using an example, the application of Z-differential equations in medicine is clarified. It is demonstrated that Z-differential equations outperform FDEs in making a decision under uncertainty.

中文翻译：

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 Z-微分方程


本文致力于建立一个框架来研究一类称为 Z 微分方程的不确定微分方程。为了达到这个目的，我们首先介绍基于半粒函数的$Z^{+}$-数的四种基本运算。然后，根据$Z^{+}$-数空间上的度量定义，给出Z-数值函数的极限和连续性概念。此外，还介绍了Z数值函数的Z-可微性、Z-积分和Z-拉普拉斯变换的概念。另外，通过给出本文证明的一些理论，建立了微积分的基础——Z-微积分。我们进一步给出了研究 Z 微分方程的存在性和唯一性的理论。还显示了 Z 微分方程和 $Z^{+}$-数字之间的概念统一。概念上的统一表明，Z 微分方程可以表示为结合模糊微分方程 (FDE) 和随机微分方程的双峰微分方程。此外，引入了称为$(s,\mu)$-cut的双峰切割的概念，并解释了它与其他新概念（例如可接受的时间和可接受的信息区域）的关系。通过一个例子，阐明了Z-微分方程在医学中的应用。结果表明，Z 微分方程在不确定性下做出决策方面优于 FDE。