On One-Sided Lipschitz Stability of Set-Valued Contractions

We give conditions under which the distance from a point x to the set of fixed points of the composition of the set-valued mappings F and G is bounded by a constant times the smallest distance between F ^?1(x) and G(x). This estimate allows us to significantly sharpen a result by T.-C. Lim [10 T.-C. Lim ( 1985 ). On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations . J. Math. Anal. Appl 110 : 436 – 441 .[Crossref], [Web of Science ®] , [Google Scholar]] regarding fixed-points stability of set-valued contractions. A global version of the Lyusternik-Graves theorem is obtained from this estimate as well. We apply the generalization of Lim's result to establish one-sided Lipschitz properties of the solution mapping of a differential inclusion with a parameter.