Transitivity, mixing and chaos for a class of set-valued mappings

Consider the continuous map →: X → X and the continuous map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar f$$ \end{document} of K(X) into itself induced by →, where X is a metric space and K(X) the space of all non-empty compact subsets of X endowed with the Hausdor. metric. According to the questions whether the chaoticity of → implies the chaoticity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar f$$ \end{document} posed by Román-Flores and when the chaoticity of → implies the chaoticity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar f$$ \end{document} posed by Fedeli, we investigate the relations between → and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar f$$ \end{document} in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Román-Flores's question and Fedeli's question.