We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.
In this paper, we study the dynamical properties of set-valued dynamical systems. Specifically, we focus on the sensitivity, transitivity and mixing of set-valued dynamical systems. Under the setting of set-valued case, we define sensitivity and investigate its properties. We also study the transitivity and mixing of set-valued dynamical systems that have been defined. We show that both transitivity and mixing are invariant under topological conjugacy. We also discuss some implication results on the product set-valued function constructed from two different set-valued functions equipped with various transitivity and mixing conditions.
In this paper, we introduce fuzzy generalized β-F-contractions as a generalization of fuzzy F-contractions with admissible mappings. We deduce sufficient conditions for the existence and uniqueness of fixed points for fuzzy generalized β-F-contractions in complete strong fuzzy metric spaces. Our results generalize several fixed-point results from the literature. We present an application of our main result.
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