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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] 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.