Advertisement

Pseudo-contractivity and Metric Regularity in Fixed Point Theory

  • Adrian PetruşelEmail author
  • Gabriela Petruşel
  • Jen-Chih Yao
Article
  • 126 Downloads

Abstract

In this paper, we will prove some fixed point results for multi-valued almost pseudo-contractions in some generalized metric spaces. Data dependence theorems and some applications to multi-valued coincidence problems are also given.

Keywords

Quasimetric space Vector-valued metric space Multi-valued operator Fixed point Metric regularity Almost pseudo-contraction Data dependence 

Mathematics Subject Classification

47H10 49J53 54H25 

Notes

Acknowledgements

The authors are grateful to the referees for their helpful comments and to the guest editors for constructive ideas. The first two authors extend their sincere thanks to Professor J.-C. Yao for supporting, by the Grant MOST 105-2115-M-039-002-MY3, the scientific visit to NSYSU Kaohsiung, Taiwan. J.-C. Yao was partially supported by the Grant MOST 105-2221-E-039-009-MY3.

References

  1. 1.
    Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Am. Math. Soc. 355, 493–517 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dontchev, A.L., Lewis, A.S.: Perturbations and metric regularity. Set-Valued Anal. 13, 417–438 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk. 55, 103–162 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ioffe, A.D.: Towards variational analysis in metric spaces: metric regularity and fixed points. Math. Progr. Ser. B 123, 241–252 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ioffe, A.D.: Variational Analysis of Regular Mappings. Springer, Cham (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  9. 9.
    Mordukhovich, B.S., Shao, Y.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. 25, 1401–1424 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1997)zbMATHGoogle Scholar
  11. 11.
    Uderzo, A.: A metric version of Milyutin theorem. Set-Valued Var. Anal. 20, 279–306 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rockafellar, R.T.: Lipschitzian properties of multifunctions. Nonlinear Anal. 9, 867–885 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dontchev, A.L., Frankowska, H.: Lyusternik–Graves theorem and fixed points. Proc. Am. Math. Soc. 139, 521534 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Beer, G., Dontchev, A.S.: The weak Ekeland variational principle and fixed points. Nonlinear Anal. 102, 91–96 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Arutyunov, A.V.: Covering mappings in metric spaces and fixed points. Dokl. Math. 76, 665–668 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Arutyunov, A.V.: The coincidence point problem for set-valued mappings and Ulam-Hyers stability. Dokl. Math. 89, 188–191 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nadler Jr., S.B.: Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bakhtin, I.A.: The contraction mapping principle in almost metric spaces. Funct. Anal. Ulianovskii Gos. Ped. Inst. 30, 26–37 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Berinde, V.: Generalized contractions in quasimetric spaces. Seminar on Fixed Point Theory 3, 3–9 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1, 5–11 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Bota, M., Molnar, A., Varga, Cs: On Ekeland’s variational principle in b-metric spaces. Fixed Point Theory 12, 21–28 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kirk, W.A., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Arutyunov, A.V., Greshnov, A.V.: Theory of \((q_1, q_2)\)-quasimetric spaces and coincidence points. Dokl. Math. 94, 434–437 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Arutyunov, A.V., Greshnov, A.V., Lokutsievskii, L.V., Storozhuk, K.V.: Topological and geometrical properties of spaces with symmetric and nonsymmetric \(f\)-quasimetrics. Topol. Its Appl. 221, 178–194 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Berinde, V., Choban, M.: Generalized distances and their associate metrics. Impact on fixed point theory. Creat. Math. Inf. 22, 23–32 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Cobzaş, Şt.: \(b\)-Metric and generalized b-metric spaces. Fixed Point Theory 19 (2018) (to appear)Google Scholar
  27. 27.
    Czerwik, S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Univ. Modena 46, 263–276 (1998)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Azé, D., Penot, J.-P.: On the dependence of fixed point sets of pseudo-contractive multifunctions. Application to differential inclusions. Nonlinear Dyn. Syst. Theory 6, 31–47 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Dontchev, A.L., Hager, W.W.: An inverse mapping theorem for set-valued maps. Proc. Am. Math. Soc. 121, 481–489 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Miculescu, R., Mihail, A.: New fixed point theorems for set-valued contractions in b-metric spaces. J. Fixed Point Theory Appl. 19, 2153–2163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, Springer Math. Monographs. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  32. 32.
    Berinde, M., Berinde, V.: On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 326, 772–782 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, T.X.: Fixed point theorems and fixed point stability for multi-valued mappings on metric spaces. Nanjing Daxue Xuebao Shuxue Bannian Kan 6, 16–23 (1989)MathSciNetzbMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Petruşel, A., Petruşel, G., Yao, J.-C.: Fixed point and coincidence point theorems in b-metric spaces with applications. Appl. Anal. Discrete Math. 11, 199–215 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Arutyunov, A.V., Greshnov, A.V.: Coincidence points of multivalued mappings in \((q_1;q_2)\)- quasimetric spaces. Dokl. Math. 96, 438–441 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Arutyunov, A.V., Avakov, E.R., Zhukovskiy, S.E.: Stability theorems for estimating the distance to a set of coincidence points. SIAM J. Optim. 25, 807–828 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Petruşel, A., Petruşel, G., Yao, J.-C.: Contributions to the coupled coincidence point problem in b-metric spaces with applications. Filomat 31, 3173–3180 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Perov, A.I.: On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Diff. Uvavn. 2, 249–264 (1964). (in Russian)MathSciNetGoogle Scholar
  40. 40.
    Varga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  41. 41.
    Arutyunov, A.V., Zhukovskiy, S.E.: Coincidence points of mappings in vector metric spaces with applications to differential equations and control systems. Differ. Equ. 53, 1440–1448 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zabrejko, P.P.: K-metric and K-normed linear spaces: survey. Collect. Math. 48, 825–859 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Babeş-Bolyai University Cluj-NapocaCluj-NapocaRomania
  2. 2.Academy of Romanian Scientists BucharestBucharestRomania
  3. 3.Center for General Education Taichung 404China Medical UniversityTaichungTaiwan

Personalised recommendations