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

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Abstract

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 \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} = \infty}\).

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References

  1. C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, 2008.

  2. Mann W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4(3), 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  3. Reich S.: Fixed point iterations of nonexpansive mappings. Pacific J. Math. 60(2), 195–198 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dotson W.G.: On the Mann iterative process. Trans. Amer. Math. Soc. 149(1), 65–73 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Reich S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67(2), 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Genel A., Lindenstrauss J.: An example concerning fixed points. Israel J. Math. 22(1), 81–86 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colao V., Marino G.: Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory and Applications 1, 39 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Song Y., Chen R.: Viscosity approximation methods for nonexpansive non-selfmappings. J. Math. Anal. Appl. 321(1), 316–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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)

    Article  MATH  Google Scholar 

  12. B.R. Halpern, Fixed point theorems for outward maps, Doctoral Thesis, Univ. Of California, Los Angeles, CA, 1965.

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Reich S.: On fixed point theorems obtained from existence theorems for differential equations. J. Math. Anal. Appl. 54(1), 26–36 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Dekker, 1984.

  16. Reich S.: Fixed points of nonexpansive functions. J. London Math. Soc. 7, 5–10 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fejér L.: Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen. Math. Ann. 85(1), 41–48 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  18. Motzkin T.S., Schoenberg I.J.: The relaxation method for linear inequalities. Canadian J. Math. 6(3), 393–404 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  19. H.H. Bauschke, P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2017.

  20. Clarkson J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40(3), 396–414 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Fréchet, Sur la notion de diff´erentielle d’une fonction de ligne. Trans. Amer. Math. Soc. (1914), 135–161.

  24. Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73(4), 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  25. Browder F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Amer. Math. Soc. 74(4), 660–665 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  36. Kopecká E., Reich S.: Nonexpansive retracts in Banach spaces. Banach Center Publications 77, 161–174 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ishikawa S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59(1), 65–71 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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  1. Dipartimento di Matematica, Universitá della Calabria, via P. Bucci, 87036, Rende (CS), Italy

    Giuseppe Marino & Luigi Muglia

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  1. Giuseppe Marino
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  2. Luigi Muglia
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Both authors contributed equally and significantly in writing the paper. Both authors read and approved the final manuscript.

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Correspondence to Giuseppe Marino.

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Marino, G., Muglia, L. Boundary Point Method and the Mann–Dotson Algorithm for Non-self Mappings in Banach Spaces. Milan J. Math. 87, 73–91 (2019). https://doi.org/10.1007/s00032-019-00293-4

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