In this paper we introduce and study semigroups of operators on spaces offuzzy-number-valued functions, and various applications to fuzzy differentialequations are presented. Starting from the space of fuzzy numbers, many newspaces sharing the same properties are introduced. We derive basic operatortheory results on these spaces and new results in the theory of semigroups oflinear operators on fuzzy-number kind spaces. The theory we develop is used tosolve classical fuzzy systems of differential equations, including, forexample, the fuzzy Cauchy problem and the fuzzy wave equation. These toolsallow us to obtain explicit solutions to fuzzy initial value problems whichbear explicit formulas similar to the crisp case, with some additional fuzzyterms which in the crisp case disappear. The semigroup method displays a clearadvantage over other methods available in the literature (i.e., the level setmethod, the differential inclusions method and other "fuzzification" methods ofthe real-valued solution) in the sense that the solutions can be easilyconstructed, and that the method can be applied to a larger class of fuzzydifferential equations that can be transformed into an abstract Cauchy problem.
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