On fixed point stability for set-valued contractive mappings with applications to generalized differential equations

where d(x, C) = inf,..,. d(x, c). A set-valued mapping T: X-+ C(X) is a cI < 1 and for x, y E X. If X is complete, then every set-valued contraction has a fixed point, i.e., a point x with x E TX. The set of fixed points of T will be denoted by F(T). Stability of fixed points of set-valued contractions was investigated in [12] and [lo]. In [lo], a stability theorem was proved under some rather restrictive conditions, including (i) the domain of the maps being a closed convex-bounded subset of a Hilbert space, (ii) the image of each point under each map being a closed convex subset. Theorem 1 below removes these conditions.