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Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces

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Abstract

In this paper, we propose a new modified proximal point algorithm for finding a common element of the set of common minimizers of a finite family of convex and lower semi-continuous functions and the set of common fixed points of a finite family of nonexpansive mappings in complete CAT(0) spaces, and prove some convergence theorems of the proposed algorithm under suitable conditions. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

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References

  1. Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Springer-Verlag, Berlin, Heidelberg (1999)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 357, 89128 (2015)

    MathSciNet  Google Scholar 

  4. Tits, J.: A Theorem of Lie-Kolchin for Trees, Contributions to Algebra: a Collection of Papers Dedicated to Ellis Kolchin. Academic Press, New York (1977)

    MATH  Google Scholar 

  5. Martinet, B.: Regularisation dinquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)

    MATH  Google Scholar 

  6. Ariza-Ruiz, D., Leustean, L., Lopez, G.: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 366, 4299–4322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bačák, M.: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689–701 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Marino, G., Xu, H. K.: Convergence of generalized proximal point algorithm. Commun. Pure. Appl. Anal. 3, 791–808 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cholamjiak, P., Abdou, A. A., Cho, Y. J.: Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 227 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dhompongsa, S., Kirk, W. A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kirk, W. A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dhompongsa, S., Kirk, W. A., Panyanak, B.: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 8, 35–45 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Dhompongsa, S., Panyanak, B.: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kopecká, E., Reich, S.: Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball. Nonlinear Anal. 70, 3187–3194 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chidume, C. E., Bello, A. U., Ndambomve, P.: Strong and Δ-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal. Article ID 805168 (2014)

  18. Guler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 403419, 29 (1991)

    MathSciNet  Google Scholar 

  19. Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math Helv. 70, 659673 (1995)

    Article  MathSciNet  Google Scholar 

  20. Mayer, U. F.: Gradient flows on nonpositively curved metric spaces and harmonic maps. Commun. Anal. Geom. 6, 199–253 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Basel, Lectures in Mathematics ETH Zrich. Birkhuser (2008)

    MATH  Google Scholar 

  22. Combettes, P. L., Pesquet, J. C.: Proximal Splitting Methods in Signal Processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, p 185212. Springer, New York (2011)

    Google Scholar 

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Acknowledgements

The authors would like to thank the referees for valuable suggestions on the research paper. This paper was supported by the Thailand Research Fund under the project RTA5780007.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand

    Kritsada Lerkchaiyaphum & Withun Phuengrattana

  2. Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand

    Withun Phuengrattana

Authors
  1. Kritsada Lerkchaiyaphum
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  2. Withun Phuengrattana
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Correspondence to Withun Phuengrattana.

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Lerkchaiyaphum, K., Phuengrattana, W. Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces. Numer Algor 77, 727–740 (2018). https://doi.org/10.1007/s11075-017-0337-6

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  • DOI: https://doi.org/10.1007/s11075-017-0337-6

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