Abstract

This work continues the studies in our previous paper ["Lyusternik-Graves theorem and fixed points", Proc. Amer. Math. Soc. 139 (2011) 521–534]. It is written as a separate paper which extends the previous one in the direction of closing the gap between Lyusternik-Graves theorems and fixed point theorems. Here we introduce a new definition of global metric regularity on a set and associated definitions of Aubin continuity and linear openness that are equivalent to metric regularity on the same sets and with the same constant. When the sets are neighborhoods of a point in the graph of the mapping, these definitions reduce to the well studied properties at a point. We present Lyusternik-Graves type theorems in metric spaces for single-valued and set-valued perturbations, and show that they can be derived from, and some of them are even equivalent to, corresponding set-valued fixed point theorems.

Author information

Contact details are reproduced from the original publication and may be historical.

Asen L. Dontchev

Mathematical Reviews and the University of Michigan, Ann Arbor, MI, U.S.A.
and: Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

ald@ams.org

Hélène Frankowska

CNRS – Institut de Mathématiques, Université P. et M. Curie, 4 place Jussieu, 75252 Paris, France

frankowska@math.jussieu.fr

Keywords

  • Set-valued analysis
  • metric regularity
  • Aubin property
  • linear openness
  • contraction mapping
  • Lyusternik-Graves theorem
  • Milyutin theorem

Mathematics Subject Classification

49J53, 47J07, 58C15, 49K40, 90C31

References

  1. A. V. Arutyunov: Covering mappings in metric spaces, and fixed points, Dokl. Akad. Nauk 416 (2007) 151–155 (in Russian); Dokl. Math. 76 (2007) 665–668 (in English).https://doi.org/10.1134/S1064562407050079Google Scholar
  2. A. V. Arutyunov, E. Avakov, B. Gel´ man, A. Dmitruk, V. Obukhovskiĭ: Locally covering maps in metric spaces and coincidence points, J. Fixed Point Theory Appl. 5 (2009) 105–127.https://doi.org/10.1007/s11784-008-0096-zGoogle Scholar
  3. J.-P. Aubin: Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984) 87–111.https://doi.org/10.1287/moor.9.1.87Google Scholar
  4. D. Azé, J.-N. Corvellec: On some variational properties of metric spaces, J. Fixed Point Theory Appl. 5 (2009) 185–200.https://doi.org/10.1007/s11784-008-0054-9Google Scholar
  5. A. V. Dmitruk, A. A. Milyutin, N. P. Osmolovskiĭ: Lyusternik’s theorem and the theory of extrema, Usp. Mat. Nauk 35(6) (216) (1980) 11–46 (in Russian); Russ. Math. Surv. 35(6) (1980) 11–51 (in English).https://doi.org/10.1070/RM1980v035n06ABEH001973Google Scholar
  6. A. V. Dmitruk, A. Kruger: Extensions of metric regularity, Optimization 58 (2009) 561–584.https://doi.org/10.1080/02331930902928674Google Scholar
  7. A. L. Dontchev, W. W. Hager: An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481–489.https://doi.org/10.1090/S0002-9939-1994-1215027-7Google Scholar
  8. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings, Springer, New York (2009).https://doi.org/10.1007/978-0-387-87821-8Google Scholar
  9. A. L. Dontchev, H. Frankowska: Lyusternik-Graves theorem and fixed points, Proc. Amer. Math. Soc. 139 (2011) 521–534.https://doi.org/10.1090/S0002-9939-2010-10490-2Google Scholar
  10. M. Durea: Openness properties for parametric set-valued mappings and implicit multifunctions, Nonlinear Anal. 72 (2010) 571–579.https://doi.org/10.1016/j.na.2009.06.105Google Scholar
  11. H. Frankowska: Some inverse mapping theorems, Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990) 183–234.https://doi.org/10.1016/s0294-1449(16)30300-6Google Scholar
  12. A. D. Ioffe: Metric regularity and subdifferential calculus, Usp. Mat. Nauk 55(3) (333) (2000) 103–162 (in Russian); Russ. Math. Surv. 55(3) (2000) 501–558 (in English).https://doi.org/10.4213/rm292https://doi.org/10.1070/RM2000v055n03ABEH000292Google Scholar
  13. A. D. Ioffe: Towards variational analysis in metric spaces: metric regularity and fixed points, Math. Program., Ser. B 123 (2010) 241–252.https://doi.org/10.1007/s10107-009-0316-3Google Scholar
  14. S. B. Nadler, Jr.: Multi-valued contraction mapping, Pac. J. Math. 30 (1969) 475–488.https://doi.org/10.2140/pjm.1969.30.475Google Scholar
  15. R. T. Rockafellar: Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985) 867–885.https://doi.org/10.1016/0362-546X(85)90024-0Google Scholar
  16. Y. Sekiguchi: Regularity in optimization problems and its quantitative analysis, in: Applied Functional Analysis (Tokyo, 2005), T. Murofushi et al. (ed.), Yokohama Publ., Yokohama (2007) 169–176.Google Scholar
  17. A. Uderzo: On some regularity properties in variational analysis, Set-Valued Var. Anal. 17 (2009) 409–430.https://doi.org/10.1007/s11228-009-0121-4Google Scholar
  18. C. Ursescu: Inherited openness, Rev. Roum. Math. Pures Appl. 41 (1996) 401–416.Google Scholar
  19. X. Y. Zheng, K. F. Ng: Metric regularity of composite multifunctions in Banach spaces, Taiwanese J. Math. 13 (2009) 1723–1735.https://doi.org/10.11650/twjm/1500405611Google Scholar

Suggested citation

A. L. Dontchev, H. Frankowska. “Lyusternik-Graves Theorem and Fixed Points II.” Journal of Convex Analysis 19 (2012), No. 4, 955–973. https://doi.org/10.68381/jca19050

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.