Boundary Point Method and the Mann–Dotson Algorithm for Non-self Mappings in Banach Spaces

  • Giuseppe MarinoEmail author
  • Luigi Muglia


Let C be a closed, convex and nonempty subset of a Banach space X. Let \({T : C \rightarrow X}\) be a nonexpansive inward mapping. We consider the boundary point map \({h_{C,T } : C \rightarrow \mathbb{R}}\) depending on C and T defined by \({h_{C,T} = {\rm max}\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}}\), for all \({x \in C}\). Then for a suitable step-by-step construction of the control coefficients by using the function \({h_{C,T }}\), we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} < \infty}\) and weak convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} = \infty}\).


Boundary point method nonexpansive mappings inward condition non-self mappings 

Mathematics Subject Classification (2010)

Primary 47H05 Secondary 47H10 65J05 


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The authors thank the anonymous reviewers for for their valuable suggestions regarding the improvement of the paper. This paper is funded by Ministero dell’Istruzione, Universitá e Ricerca (MIUR) and Gruppo Nazionale di Analisi Matemarica e Probabilitá e Applicazioni (GNAMPA).

The first author declares that his contribution would not have been possible without the help of Maria Grazia Atzeri, Carla Mazzone and of the co-author Luigi Muglia.

Author’s contribution

Both authors contributed equally and significantly in writing the paper. Both authors read and approved the final manuscript.

Competing interest

The author declare that they have no competing interest.


  1. 1.
    C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, 2008.Google Scholar
  2. 2.
    Mann W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4(3), 506–510 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Reich S.: Fixed point iterations of nonexpansive mappings. Pacific J. Math. 60(2), 195–198 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dotson W.G.: On the Mann iterative process. Trans. Amer. Math. Soc. 149(1), 65–73 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Reich S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67(2), 274–276 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Genel A., Lindenstrauss J.: An example concerning fixed points. Israel J. Math. 22(1), 81–86 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauschke H.H., Matoušková E., Reich S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. TMA 56(5), 715–738 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colao V., Marino G.: Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory and Applications 1, 39 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Song Y., Chen R.: Viscosity approximation methods for nonexpansive non-selfmappings. J. Math. Anal. Appl. 321(1), 316–326 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y.J. Cho, J.I. Kang, X. Qin, M. Shang, Weak and strong convergence theorems of a Mann-type iterative algorithm for k-strict pseudo-contractions, Taiwanese J. Math. (2010), 1439–1455.Google Scholar
  11. 11.
    Bo L.H., Yi L.: Strong convergence theorems of the Halpern-Mann’s mixed iteration for a totally quasi-\({\phi}\)-asymptotically nonexpansive non-self multi-valued mapping in Banach spaces. Journal of Inequalities and Applications 2014(1), 225 (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    B.R. Halpern, Fixed point theorems for outward maps, Doctoral Thesis, Univ. Of California, Los Angeles, CA, 1965.Google Scholar
  13. 13.
    Halpern B.R., Bergman G.M.: A fixed-point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130(2), 353–358 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Reich S.: On fixed point theorems obtained from existence theorems for differential equations. J. Math. Anal. Appl. 54(1), 26–36 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Dekker, 1984.Google Scholar
  16. 16.
    Reich S.: Fixed points of nonexpansive functions. J. London Math. Soc. 7, 5–10 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fejér L.: Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen. Math. Ann. 85(1), 41–48 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Motzkin T.S., Schoenberg I.J.: The relaxation method for linear inequalities. Canadian J. Math. 6(3), 393–404 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    H.H. Bauschke, P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2017.Google Scholar
  20. 20.
    Clarkson J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40(3), 396–414 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bruck R.E.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. 32(2–3), 107–116 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tan K.K., Xu H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. Fréchet, Sur la notion de diff´erentielle d’une fonction de ligne. Trans. Amer. Math. Soc. (1914), 135–161.Google Scholar
  24. 24.
    Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73(4), 591–597 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Browder F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Amer. Math. Soc. 74(4), 660–665 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    S. He, W. Zhu, A modified Mann iteration by boundary point method for finding minimum-norm fixed point of nonexpansive mappings, Abstract and Applied Analysis 2013, Hindawi.Google Scholar
  27. 27.
    He S., Yang C.: Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory and Applications 2014(1), 56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tufa A.R., Zegeye H.: Mann- and Ishikawa-type iterative schemes for approximating fixed points of multi-valued non-self mappings, Mediterranean J. Math. 13(6), 4369–4384 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tufa A.R., Zegeye H.: Krasnoselskii–Mann method for multi-valued mon-Self mappings in CAT(0) Spaces. Filomat 31(14), 4629–4640 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tufa A.R., Zegeye H., Thuto M.: Convergence theorems for non-self mappings in CAT(0) spaces. Numer. Functional Anal. Opt. 38(6), 705–722 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Colao V., Marino G., Hussain N.: On the approximation of fixed points of non-self strict pseudocontractions. Rev. R. Acad. Cienc. Exactas, Fs. Nat. Ser A. Math. 111(1), 159–165 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Takele M.H., Reddy B.K.: Fixed Point Theorems for Approximating a Common Fixed Point for a Family of non-self, Strictly Pseudo Contractive and Inward Mappings in Real Hilbert Spaces. Global J. Pure Appl. Math. 13(7), 3657–3677 (2017)Google Scholar
  33. 33.
    Guo M., Li X., Su Y.: On an open question of V. Colao and G. Marino presented in the paper: Krasnoselskii–Mann method for non-self mappings. SpringerPlus 5(1), 1328 (2016)CrossRefGoogle Scholar
  34. 34.
    Colao V., Marino G., Muglia L.: On the Approximation of Zeros of Non-Self Monotone Operators. Numer. Functional Anal. Opt. 37(6), 667–679 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Baillon J.-B., Bruck R.E., Reich S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4, 1–9 (1978)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kopecká E., Reich S.: Nonexpansive retracts in Banach spaces. Banach Center Publications 77, 161–174 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Ishikawa S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59(1), 65–71 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá della CalabriaRende (CS)Italy

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