The graph topology of set-valued mappings and the generic stability of fixed points (Q2739177)

The authors introduce the graph topology. They use this topology to study further the stability of fixed points and to prove that the set of fixed points of a set-valued mapping has generic stability. They show that for most mappings all fixed points are essential. Theorems 3, 4 and 5 gave the essential fixed point with respect to the graph topology, these results are quite different from the existing results on the essential fixed points, because the graph topology is much weaker than the topology induced by the uniform metric. Therefore, the results in this paper are more general than the corresponding results with respect to the topology induced by the uniform metric. These results will have some important applications to generic stability.