On One-Sided Lipschitz Stability of Set-Valued Contractions

Abstract

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.

Keywords:

• Composition
• Differential inclusions
• Fixed points
• Lyusternik-Graves theorem
• Lipschitz stability
• Set-valued mappings

Mathematics Subject Classification:

• 49J53
• 47H10
• 49K40
• 90C31

ACKNOWLEDGMENTS

The authors wish to thank the anonymous referees for their valuable comments and suggestions.

Part of the special issue, “Variational Analysis and Applications.”

Notes

Following [7], we require that Lipschitz continuous mappings must be closed valued to avoid the situation when Lipschitz continuity does not entail continuity. This is also assumed implicitly in [10].