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Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory

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Abstract

In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at generalized compositions. Then we obtain, in purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks.

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References

  1. Adly, S., Dontchev, A.L., Théra, M.: On one-sided Lipschitz stability of set-valued contractions. Numer. Funct. Anal. Optim. 35, 837–850 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apetrii, M., Durea, M., Strugariu, R.: On subregularity properties of set-valued mappings. Applications to solid vector optimization. Set-Valued and Variational Analysis 21, 93–126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arutyunov, A.V.: Covering mapping in metric spaces, and fixed points. Dokl. Math. 76, 665–668 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arutyunov, A.V: Stability of coincidence points and properties of covering mappings. Mathematical Notes 86, 153–158 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arutyunov, A., Avakov, E., Gel’man, B., Dmitruk, A., Obukhovskii, V.: Locally covering maps in metric spaces and coincidence points. J. fixed point theory appl. 5, 105–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arutyunov, A.V., Avakov, E.R., Izmailov, A.F.: Directional regularity and metric regularity. SIAM J. Optim. 18, 810–833 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arutyunov, A.V., Avakov, E.R., Zhukovskiy, S.E.: Stability theorems for estimating the distance to a set of coincidence points, SIAM Journal on Optimization, to appear

  8. Arutyunov, A.V., Zhukovskiy, S.E.: Perturbation of solutions of the coincidence point problem for two mappings. Dokl. Math. 89, 346–348 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dmitruk, A.V.: On a nonlocal metric regularity of nonlinear operators. Control. Cybern. 34, 723–746 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Dmitruk, A.V., Kruger, A.Y.: Metric regularity and systems of generalized equations. Aust. J. Math. Anal. Appl. 342, 864–873 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dmitruk, A.V., Kruger, A.Y.: Extensions of metric regularity. Optimization 58, 561–584 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dmitruk, A.V., Milyutin, A.A., Osmolovskii, N.P.: Lyusternik’s theorem and the theory of extrema. Uspekhi Mat Nauk 35, 11–46 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Dontchev, A.L.: The Graves theorem revisited. Journal of Convex Analysis 3, 45–53 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points. Proc. Am. Math. Soc. 139, 521–534 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points II. Journal of Convex Analysis 19, 955–973 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  17. Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems: Applications to variational systems, Nonlinear Analysis, Theory. Math. Methods Appl. 75, 1246–1259 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Durea, M., Strugariu, R.: Chain rules for linear openness in general Banach spaces. SIAM J. Optim. 22, 899–913 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Durea, M., Strugariu, R.: Chain rules for linear openness in metric spaces and applications, Mathematical Programming. Ser. A 143, 147–176 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued and Variational Analysis 21, 151–176 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gfrerer, H.: On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs. Set-Valued and Variational Analysis 22, 79–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ioffe, A.D: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55(3(333)),103–162 (2000); English translation in Mathematical Surveys, 55, 501–558 (2000)

  23. Ioffe, A.D.: Towards variational analysis in metric spaces: metric regularity and fixed points, Mathematical Programming. Ser. B 123, 241–252 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Ioffe, A.D: Regularity on a fixed set. SIAM J. Optim. 21, 1345–1370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64, 49–79 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lim, T.-C.: On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations. J. Math. Anal. Appl. 110, 436–441 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ngai, H.V., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19, 1–20 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ngai, H.V., Théra, M.: Directional metric regularity of multifunctions. available at arXiv:1304.7748

  29. Uderzo, A.: A metric version of Milyutin Theorem. Set-Valued and Variational Analysis 20, 279–306 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ursescu, C.: Inherited openness. Revue Roumaine des Mathématiques Pures et Appliquées 41(5–6), 401–416 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Bd. Carol I, nr. 11, 700506, Iaşi, Romania

    Marius Durea

  2. Department of Mathematics and Informatics, “Gheorghe Asachi” Technical University of Iaşi, Bd. Carol I, nr. 11, 700506, Iaşi, Romania

    Radu Strugariu

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  1. Marius Durea
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  2. Radu Strugariu
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Correspondence to Marius Durea.

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Durea, M., Strugariu, R. Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory. Set-Valued Var. Anal 24, 231–251 (2016). https://doi.org/10.1007/s11228-015-0327-6

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  • DOI: https://doi.org/10.1007/s11228-015-0327-6

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