Fixed Points of Pseudo-Contractive Holomorphic Mappings

Chapter
First Online:
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

Abstract

We study conditions that ensure the existence of fixed points of pseudo-contractive mappings originally considered by Browder, Kato, Kirk and Morales. Specifically we consider holomorphic pseudo-contractions on the open unit ball of a complex Banach space which in general are not necessarily bounded. As a consequence, we obtain sufficient conditions for the existence and uniqueness of the common fixed point of a semigroup of holomorphic self-mappings and study its rate of convergence to this point.

Keywords

Holomorphic maps Fixed point theory Contractive maps 

Mathematics Subject Classification

37C25 32A10 
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Notes

Acknowledgements

The work was partially supported by the European Commission under the project STREVCOMS PIRSES-2013-612669. The publication was prepared with the support of the “RUDN University Program 5-100”. Both authors are grateful to the anonymous referee for the very fruitful remarks.

References

  1. 1.
    Aharonov, D., Elin, M., Reich, S., Shoikhet, D.: Parametric representations of semi-complete vector fields on the unit balls in \({\mathbb C}^n\) and in Hilbert space. Atti Accad. Naz. Lincei 10, 229–253 (1999)Google Scholar
  2. 2.
    Browder, F.E.: Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 73, 875–882 (1967)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Budzyńska, M., Kuczumow, T., Reich, S.: The Denjoy-Wolff iteration property in complex Banach spaces. J. Nonlinear Convex Anal. 17, 1213–1221 (2016)MathSciNetMATHGoogle Scholar
  4. 4.
    Dineen, S.: The Schwartz Lemma. Clarendon, Oxford (1989)MATHGoogle Scholar
  5. 5.
    Earle, C.J., Hamilton, R.S.: A fixed point theorem for holomorphic mappings. In: Proceedings of Symposia in Pure Mathematics, vol. 16, pp. 61–65. American Mathematical Society, Providence, RI (1970)Google Scholar
  6. 6.
    Elin, M.: Extension operators via semigroups. J. Math. Anal. Appl. 377, 239–250 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. Marcel Dekker, New York (1984)Google Scholar
  8. 8.
    Graham, I., Kohr, G.: Univalent mappings associated with the Roper-Suffridge extension operator. J. Anal. Math. 81, 331–342 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Harris, L.A.: Schwarz-Pick systems of pseudometrics for domains in normed linear spaces. In: Advances in Holomorphy, pp. 345–406. North Holland, Amsterdam (1979)Google Scholar
  10. 10.
    Harris, L.A.: Fixed points of holomorphic mappings for domains in Banach spaces. Abstr. Appl. Anal. 5, 261–274 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Harris, L.A.: Fixed point theorems for infinite dimensional holomorphic functions. J. Korean Math. Soc. 41, 175–192 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Khatskevich, V., Reich, S., Shoikhet, D.: Complex dynamical systems on bounded symmetric domains. Electron. J. Differ. Equ. 19, 1–9 (1997)MathSciNetMATHGoogle Scholar
  13. 13.
    Kirk, W.A., Schöneberg, R.: Some results on pseudocontractive mappings. Pac. J. Math. 71, 89–100 (1977)CrossRefMATHGoogle Scholar
  14. 14.
    Krasnosel’skiï, M.A., Zabreïko, P.P.: Geometrical Methods of Nonlinear Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 263. Springer, Berlin (1984)Google Scholar
  15. 15.
    Krasnosel’skiï, M.A., Vaïnikko, G.M., Zabreïko, P.P., Rutitskii, Ya.B., Stecenko, V.Ya.: Approximate Solution of Operator Equations. Wolters-Noordhoff, Groningen (1972)Google Scholar
  16. 16.
    Kuczumow, T., Reich, S., Shoikhet, D.: Fixed points of holomorphic mappings: a metric approach. In: Handbook of Metric Fixed Point Theory, pp. 437–515. Kluwer, Dordrecht (2001)Google Scholar
  17. 17.
    MacCluer, B.D.: Iterates of holomorphic self-maps of the unit ball in \(\mathbb {C}^n\). Michigan Math. J. 30, 97–106 (1983)Google Scholar
  18. 18.
    Morales, C.: Pseudo-contractive mappings and the Leray-Schauder boundary condition. Comment. Math. Univ. Carol. 4, 745–756 (1979)MATHGoogle Scholar
  19. 19.
    Morales, C.: Remarks on pseudo-contractive mappings. J. Math. Anal. Appl. 87, 158–164 (1982)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pfaltzgraff, J.A., Suffridge, T.J.: An extension theorem and linear invariant families generated by starlike maps. Ann. Univ. Mariae Curie-Sk lodowska Sect. A 53, 193–207 (1999)MathSciNetMATHGoogle Scholar
  21. 21.
    Reich, S.: Minimal displacement of points under weakly inward pseudo-lipschitzian mappings. Atti Accad. Naz. Lincei 59, 40–44 (1975)MathSciNetMATHGoogle Scholar
  22. 22.
    Reich, S.: On the fixed point theorems obtained from existence theorems for differential equations. J. Math. Anal. Appl. 54, 26–36 (1976)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Reich, S., Shoikhet, D.: Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstr. Appl. Anal. 1, 1–44 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Reich, S., Shoikhet, D.: Semigroups and generators on convex domains with the hyperbolic metric. Atti Accad. Naz. Lincei 8, 231–250 (1997)MathSciNetMATHGoogle Scholar
  25. 25.
    Reich, S., Shoikhet, D.: The Denjoy-Wolff theorem. Ann. Univ. Mariae Curie-Sk lodowska Sect. A 51, 219–240 (1997)MathSciNetMATHGoogle Scholar
  26. 26.
    Reich, S., Shoikhet, D.: Averages of holomorphic mappings and holomorphic retractions on convex hyperbolic domains. Stud. Math. 130, 231–244 (1998)MathSciNetMATHGoogle Scholar
  27. 27.
    Reich, S., Shoikhet, D.: The Denjoy–Wolff theorem. Encyclopedia of Mathematics, Supplement 3, pp. 121–123. Kluwer, Dordrecht (2002)Google Scholar
  28. 28.
    Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and the Geometry of Domains in Banach Spaces. Imperial College, London (2005)CrossRefMATHGoogle Scholar
  29. 29.
    Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  30. 30.
    Shoikhet, D.: Semigroups in Geometrical Function Theory. Kluwer, Dordrecht (2001)CrossRefMATHGoogle Scholar
  31. 31.
    Stachura, A.: Iterates of holomorphic self-maps of the unit ball in Hilbert space. Proc. Am. Math. Soc. 93, 88–90 (1985)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Trenogin, V.A.: Functional Analysis. Nauka, Moscow (1980)MATHGoogle Scholar
  33. 33.
    Vainberg, M.M., Trenogin, V.A.: Theory of Bifurcation of Solutions of Nonlinear Equations. Nauka, Moscow (1969)MATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsKarmielIsrael
  2. 2.Department of MathematicsHolonIsrael

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