Continuous dependence of fixed point sets

The stability of the fixed point sets of a uniformly convergent sequence of set valued contractions is proved under the assumption that the maps are defined on a closed bounded subset B of Hubert space and take values in the family of nonempty closed convex subsets of B. In (1) the convergence of a sequence of fixed points of a convergent sequence of set valued contractions was investigated in a metric space setting. By restricting the underlying space to be a Hubert space we prove the convergence of the sequence of fixed point sets of a convergent sequence of set valued contractions. This also extends a similar result for point valued maps (2, Theorem (10.1.1)) to the set valued case. Let A be a closed bounded subset of a Hubert space H, d the norm of H, and D the Hausdorff metric on the closed subsets of A generated by d. We assume that the family of set valued maps Fk, k=0, I, ■ ■ ■ , satisfy