Topological stability in set-valued dynamics

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [ 14 ]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [ 15 ] and satisfies the positive pseudo-orbit tracing property [ 9 ] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [ 1 ] and [ 14 ].