On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis

Abstract

In this paper we establish an estimate for the rate of convergence of the Krasnosel’skiĭ-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffdingtype inequality to majorize the expected value of a convex function of these sums using Poisson distributions.

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References

  1. [1]

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1965.

    Google Scholar 

  2. [2]

    J. B. Baillon and R. E. Bruck, Optimal rates of asymptotic regularity for averaged nonexpansive mappings, in Proceedings of the Second International Conference on Fixed Point Theory and Applications (K. K. Tan, ed.), World Scientific Press, London, 1992, pp. 27–66.

    Google Scholar 

  3. [3]

    J. B. Baillon and R. E. Bruck, The rate of asymptotic regularity is \(O(1/\sqrt n )\), in Theory and Applications of Nonlinear Operators of Accretive and Monotone Types, Lecture Notes in Pure and Applied Mathematics, Vol. 178, Dekker, New York, 1996, pp. 51–81.

    Google Scholar 

  4. [4]

    J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of non-expansive mappings and semigroups in Banach spaces, Houston Journal of Mathematics 4 (1978), 1–9.

    MathSciNet  Google Scholar 

  5. [5]

    H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Analysis 56 (2004), 715–738.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    J. Borwein, S. Reich and I. Shafrir, Krasnosel’skiĭ-Mann iterations in normed spaces, Canadian Mathematical Bulletin 35 (1992), 21–28.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America 54 (1965), 1041–1044.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bulletin of the American Mathematical Society 72 (1966), 571–575.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    M. Edelstein, A remark on a theorem of M. A. Krasnosel’skiĭ, The American Mathematical Monthly 73 (1966), 509–501.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    M. Edelstein and R. C. O’Brien, Nonexpansive mappings, asymptotic regularity and successive approximations, Journal of the London Mathematical Society 17 (1978), 547–554.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn., John Wiley & Sons, New York, 1950.

    MATH  Google Scholar 

  12. [12]

    A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel Journal of Mathematics 22 (1975), 81–86.

    Article  MATH  MathSciNet  Google Scholar 

  13. [13]

    K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, in Topological Methods in Nonlinear Functional Analysis, Contemporary Mathematics, Vol. 21, American Mathematical Society, Providence, RI, 1983, pp. 115–123.

    Chapter  Google Scholar 

  14. [14]

    D. Göhde, Zum prinzip der kontraktiven Abbildung, Mathematische Nachrichten 30 (1965), 251–258.

    Article  MATH  MathSciNet  Google Scholar 

  15. [15]

    C. W. Groetsch, A note on segmenting Mann iterates, Journal of Mathematical Analysis and Applications 40 (1972), 369–372.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    W. Hoeffding, On the distribution of the number of successes in independent trials, Annals of Mathematical Statistics 27 (1956), 713–721.

    Article  MATH  MathSciNet  Google Scholar 

  17. [17]

    S. Ishikawa, Fixed points and iterations of a nonexpansive mapping in a Banach space, Proceedings of the American Mathematical Society 59 (1976), 65–71.

    Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    W. A. Kirk, A fixed point theorem for mappings which do not increase distances, American Mathematical Monthly 72 (1965), 1004–1006.

    Article  MATH  MathSciNet  Google Scholar 

  19. [19]

    W. A. Kirk, Nonexpansive mappings and asymptotic regularity, Nonlinear Analysis 40 (2000), 323–332.

    Article  MATH  MathSciNet  Google Scholar 

  20. [20]

    U. Kohlenbach, A quantitative version of a theorem due to Borwein-Reich-Shafrir, Numerical Functional Analysis and Optimization 22 (2001), 641–656.

    Article  MATH  MathSciNet  Google Scholar 

  21. [21]

    U. Kohlenbach, Uniform asymptotic regularity for Mann iterates, Journal of Mathematical Analysis and Applications 279 (2003), 531–544.

    Article  MATH  MathSciNet  Google Scholar 

  22. [22]

    M. A. Krasnosel’skiĭ, Two remarks on the method of successive approximations, Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 10 (1955), 123–127.

    MathSciNet  Google Scholar 

  23. [23]

    W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society 4 (1953), 506–510.

    Article  MATH  MathSciNet  Google Scholar 

  24. [24]

    S. Reich, Fixed point iterations of non expansive mappings, Pacific Journal of Mathematics 60 (1975), 195–198.

    Article  MATH  MathSciNet  Google Scholar 

  25. [25]

    S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 67 (1979), 274–276.

    Article  MATH  MathSciNet  Google Scholar 

  26. [26]

    S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Analysis 15 (1990), 537–558.

    Article  MATH  MathSciNet  Google Scholar 

  27. [27]

    S. Reich and A. J. Zaslavski, Convergence of Krasnosel’skiĭ-Mann iterations of nonexpansive operators, Mathematical and Computer Modelling 32 (2000), 1423–1431.

    Article  MATH  MathSciNet  Google Scholar 

  28. [28]

    H. Schaefer, Über die Methode sukzessiver Approximationen, Jahresbericht der Deutschen Mathematiker-Vereinigung 59 (1957), 131–140.

    MATH  MathSciNet  Google Scholar 

  29. [29]

    V. K. Thiruvenkatachar and T. S. Nagundiah, Inequalities concerning Bessel functions and orthogonal polynomials, Proceedings of the Indian Academy of Sciences. Section A 33 (1951), 373–384.

    MATH  Google Scholar 

  30. [30]

    J. Vaisman, Convergencia fuerte del método de medias sucesivas para operadores lineales no-expansivos, Memoria de Ingeniería Civil Matemática, Universidad de Chile, 2005.

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Correspondence to R. Cominetti.

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Supported by Fondecyt 1100046 and Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F.

Supported by Basal-Conicyt project and Núcleo Milenio Información y Coordinaci ón en Redes ICM/FIC P10-024F.

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Cominetti, R., Soto, J.A. & Vaisman, J. On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis. Isr. J. Math. 199, 757–772 (2014). https://doi.org/10.1007/s11856-013-0045-4

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Keywords

  • Banach Space
  • Nonexpansive Mapping
  • Success Probability
  • Mann Iterate
  • Catalan Number