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Generic Existence and Approximation of Fixed Points for Nonexpansive Set-valued Maps

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Abstract

We study nonexpansive set-valued maps in Banach and metric spaces. We are concerned, in particular, with the generic existence and approximation of fixed points, as well as with the structure of fixed point sets.

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References

  1. Bauschke, H.H., Borwein, J.M., Lewis, A.S.: The method of cyclic projections for closed convex sets in Hilbert space. Recent developments in optimization theory and nonlinear analysis. Contemp. Math. 204, 1–38 (1997)

    MathSciNet  Google Scholar 

  2. de Blasi, F.S.: Generic properties of some classes of operator equations. J. London Math. Soc. 23, 321–328 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. de Blasi, F.S., Myjak, J.: Sur la convergence des approximations successives pour les contractions non linéaires dans un espace de Banach. C. R. Acad. Sci. Paris 283, 185–187 (1976)

    MATH  Google Scholar 

  4. de Blasi, F.S., Myjak, J.: Sur la porosité de l’ensemble des contractions sans point fixe. C. R. Acad. Sci. Paris 308, 51–54 (1989)

    MATH  Google Scholar 

  5. Caristi, J.: Fixed point theorems for maps satsifying inwardness conditions. Trans. Amer. Math. Soc. 215, 241–251 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  7. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  8. Kirk, W.A.: Contraction mappings and extensions. In: Handbook of Metric Fixed Point Theory, pp. 1–34. Kluwer, Dordrecht (2001)

    Google Scholar 

  9. Nadler, S.B. Jr.: Multi-valued contraction mappings. Pacific J. Math. 30, 475–488 (1969)

    MATH  MathSciNet  Google Scholar 

  10. Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  11. Reich, S., Zaslavski, A.J.: Convergence of iterates of nonexpansive set-valued mappings. In: Set Valued Mappings with Applications in Nonlinear Analysis, pp. 411–420. Taylor and Francis, London (2002)

    Google Scholar 

  12. Reich, S., Zaslavski, A.J.: Generic existence of fixed points for set-valued mappings. Set-Valued Anal. 10, 287–296 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Reich, S., Zaslavski, A.J.: Two results on fixed points of set-valued nonexpansive mappings. Rev. Roumaine. Math. Pures Appl. 51, 89–94 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Ricceri, B.: Une propriété topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes. Atti Accad. Naz. Lincei 81, 283–286 (1987)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Simeon Reich.

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de Blasi, F.S., Myjak, J., Reich, S. et al. Generic Existence and Approximation of Fixed Points for Nonexpansive Set-valued Maps. Set-Valued Anal 17, 97–112 (2009). https://doi.org/10.1007/s11228-009-0104-5

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  • DOI: https://doi.org/10.1007/s11228-009-0104-5

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