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The viscosity approximation method for multivalued G-nonexpansive mappings in Hadamard spaces endowed with graphs

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Abstract

Using the viscosity approximation method introduced by Moudafi (J Math Anal Appl 241:46–55, 2000), we can obtain strong convergence theorems for monotone increasing G-nonexpansive mappings in Hadamard spaces endowed with graphs. We also give sufficient conditions for the existence of solutions of the variational inequality problem in this setting. Our results generalize and improve many results in the literature.

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References

  1. Alfuraidan, M.R., Khamsi, M.A.: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl. 2015(44), 1–10 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Anakkamatee, W., Tongnoi, B.: Convergence theorems for \(G-\)nonexpansive mappings in CAT(0) spaces endowed with graphs. Thai J. Math. 17, 775–787 (2019)

    MathSciNet  Google Scholar 

  3. Balog, L., Berinde, V.: Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph. Carpathian J. Math. 32, 293–302 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Beg, I., Butt, A.R., Radojevic, S.: The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 60, 1214–1219 (2010)

    Article  MathSciNet  Google Scholar 

  5. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 133, 195–218 (2008)

    Article  Google Scholar 

  6. Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)

    Book  Google Scholar 

  7. Brown, K.S.: Buildings. Springer, New York (1989)

    Book  Google Scholar 

  8. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)

    Article  Google Scholar 

  9. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. In: Graduate Studies in Math, vol. 33. Amer. Math. Soc., Providence (2001)

  10. Chifu, C., Karapinar, E., Petrusel, G.: Qualitative properties of the solution of a system of operator inclusions in \(b\)-metric spaces endowed with a graph. Bull. Iran. Math. Soc. 44, 1267–1281 (2018)

    Article  MathSciNet  Google Scholar 

  11. Cholamjiak, W., Suantai, S., Suparatulatorn, R., Kesornprom, S., Cholamjiak, P.: Viscosity approximation methods for fixed point problems in Hilbert spaces endowed with graphs. J. Appl. Numer. Optim. 1, 25–38 (2019)

    Google Scholar 

  12. Cuntavepanit, A., Panyanak, B.: Strong convergence of modified Halpern iterations in CAT(0) spaces. Fixed Point Theory Appl. 2011, 1–11 (2011) (Article ID 869458)

  13. Dehghan, H., Rooin, J.: Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces. J. Nonlinear Covex Anal. (accepted)

  14. Dhompongsa, S., Kaewkhao, A., Panyanak, B.: Browder’s convergence theorem for multivalued mappings without endpoint condition. Topol. Appl. 159, 2757–2763 (2012)

    Article  MathSciNet  Google Scholar 

  15. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. Marcel Dekker, New York (1984)

  19. Hunde, T.W., Sangago, M.G., Zegeye, H.: Approximation of a common fixed point of a family of \(G\)-nonexpansive mappings in Banach spaces with a graph. Int. J. Adv. Math. 2017, 137–152 (2017)

    Google Scholar 

  20. Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2008)

    Article  MathSciNet  Google Scholar 

  21. Kakavandi, B.A.: Weak topologies in complete CAT(0) metric spaces. Proc. Am. Math. Soc. 141, 1029–1039 (2013)

    Article  MathSciNet  Google Scholar 

  22. Kangtunyakarn, A.: The variational inequality problem in Hilbert spaces endowed with graphs. J. Fixed Point Theory Appl. 22(4), 1–16 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Karapinar, E.: Fixed point theorems in uniform space endowed with graph. Miskolc Math. Notes 18, 57–69 (2017)

    Article  MathSciNet  Google Scholar 

  24. Kirk, W.A.: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), pp. 195–225, Colecc. Abierta, 64, Univ. Sevilla Secr. Publ., Seville (2003)

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

    Article  MathSciNet  Google Scholar 

  26. Kopecká, E., Reich, S.: Approximating fixed points in the Hilbert ball. J. Nonlinear Convex Anal. 15, 819–829 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Marino, G., Scardamaglia, B., Karapinar, E.: Strong convergence theorem for strict pseudo-contractions in Hilbert spaces. J. Inequal. Appl. 2016(134), 1–12 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  Google Scholar 

  29. Niwongsa, Y., Panyanak, B.: Noor iterations for asymptotically nonexpansive mappings in CAT(0) spaces. Int. J. Math. Anal. 4, 645–656 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Panyanak, B.: Fixed points of multivalued \(G-\)nonexpansive mappings in Hadamard spaces endowed with graphs. J. Funct. Sp. 2020, 1–8 (2020). (Article ID 5849262)

  31. Panyanak, B., Suantai, S.: Viscosity approximation methods for multivalued nonexpansive mappings in geodesic spaces. Fixed Point Theory Appl. 2015(114), 1–14 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Reich, S., Salinas, Z.: Weak convergence of infinite products of operators in Hadamard spaces. Rend. Circ. Mat. Palermo 65, 55–71 (2016)

    Article  MathSciNet  Google Scholar 

  33. Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)

    Article  MathSciNet  Google Scholar 

  34. Shahzad, N., Markin, J.: Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces. J. Math. Anal. Appl. 337, 1457–1464 (2008)

    Article  MathSciNet  Google Scholar 

  35. Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997)

    Article  MathSciNet  Google Scholar 

  36. Thianwan, T., Yambangwai, D.: Convergence analysis for a new two-step iteration process for \(G-\)nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl. 21(44), 1–16 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Tiammee, J., Kaewkhao, A., Suantai, S.: On Browder’s convergence theorem and Halpern iteration process for \(G\)-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl. 2015(187), 1–12 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Wangkeeree, R., Preechasilp, P.: Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces. J. Inequal. Appl. 2013(93), 1–15 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Wen, D.J.: Weak and strong convergence theorems of \(G\)-monotone nonexpansive mapping in Banach spaces with a graph. Numer. Funct. Anal. Optim. 40, 163–177 (2019)

    Article  MathSciNet  Google Scholar 

  40. Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Professor Attapol Kaewkhao for a useful discussion. This research was supported by Chiang Mai University (Grant no. CMU2564).

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  1. Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Bancha Panyanak

  2. Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Bancha Panyanak

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Correspondence to Bancha Panyanak.

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Panyanak, B. The viscosity approximation method for multivalued G-nonexpansive mappings in Hadamard spaces endowed with graphs. J. Fixed Point Theory Appl. 22, 90 (2020). https://doi.org/10.1007/s11784-020-00829-x

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