模糊数值函数空间上的算子半群及其在模糊微分方程中的应用
Semigroups of Operators on Spaces of Fuzzy-Number-Valued Functions with Applications to Fuzzy Differential Equations
摘要：
本文引入并研究了模糊数值函数空间上的算子半群，并给出了它在模糊微分方程中的各种应用。从模糊数空间出发，引入了许多具有相同性质的新空间。我们得到了这些空间上的基本算子理论结果和模糊数类空间上的线性算子半群理论的新结果。我们开发的理论是用来解决经典的模糊系统的微分方程，包括，例如，模糊柯西问题和模糊波动方程。这些工具使我们能够获得明确的解决方案，模糊初值问题承担明确的公式类似于清晰的情况下，与一些额外的模糊条款，在清晰的情况下消失。半群方法显示出明显优于文献中可用的其他方法（即，水平集方法、微分包含方法和实值解的其他“模糊化”方法），在这个意义上，解可以容易地构造，并且该方法可以应用于更大类的模糊微分方程，这些模糊微分方程可以转化为抽象的柯西问题。
英文摘要：
In this paper we introduce and study semigroups of operators on spaces of fuzzy-number-valued functions, and various applications to fuzzy differential equations are presented. Starting from the space of fuzzy numbers, many new spaces sharing the same properties are introduced. We derive basic operator theory results on these spaces and new results in the theory of semigroups of linear operators on fuzzy-number kind spaces. The theory we develop is used to solve classical fuzzy systems of differential equations, including, for example, the fuzzy Cauchy problem and the fuzzy wave equation. These tools allow us to obtain explicit solutions to fuzzy initial value problems which bear explicit formulas similar to the crisp case, with some additional fuzzy terms which in the crisp case disappear. The semigroup method displays a clear advantage over other methods available in the literature (i.e., the level set method, the differential inclusions method and other "fuzzification" methods of the real-valued solution) in the sense that the solutions can be easily constructed, and that the method can be applied to a larger class of fuzzy differential equations that can be transformed into an abstract Cauchy problem.
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