Let \((X,\mathcal {U})\) be a Hausdorff uniform space and \(f_{0,\infty }=\{f_n\}_{n=0}^{\infty }\) be a sequence of uniformly continuous self-maps on X. The nonautonomous dynamical system \((X,f_{0,\infty })\) induces the set-valued dynamical system \((\mathcal {K}(X),\bar{f}_{0,\infty })\) on the hyperspace \(\mathcal {K}(X)\) consisting of all the nonempty compact subsets of X. In this paper, we mainly investigate the connections between some dynamical properties of \((X,f_{0,\infty })\) and those of \((\mathcal {K}(X),\bar{f}_{0,\infty })\) . We prove that chain mixing, shadowing property, h-shadowing property, specification property and multi- \(\mathscr {F}\) -sensitivity of \((X,f_{0,\infty })\) is equivalent to that of \((\mathcal {K}(X),\bar{f}_{0,\infty })\) , respectively. In particular, we show that chain mixing of \((X,f_{0,\infty })\) and topological mixing of \((\mathcal {K}(X),\bar{f}_{0,\infty })\) are equivalent provided that \((X,f_{0,\infty })\) has shadowing property. We obtain that positive topological entropy of \((X,f_{0,\infty })\) implies infinite entropy of \((\mathcal {K}(X),\bar{f}_{0,\infty })\) and confirm that topological equi-conjugacy between two dynamical systems is preserved by their induced set-valued systems.
