Godwin Chidi Ugwunnadi     (Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini; Department of Mathematics and Applied Mathematics, Sefako Makgato Health Science University, P.O. Box 94, Pretoria 0204, South Africa, Swaziland)


In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.


Proximal point algorithm, Total asymptotically nonexpansive mapping, Fixed point, $\triangle$ convergence, Strong convergence, CAT(0) space

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  1. Alber Ya. I., Chidume C. E., Zegeye H. Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl., 2006. Art. no. 10673. P. 1–20. DOI: 10.1155/FPTA/2006/10673
  2. Ahmad I., Ahmad M. An implicit viscosity technique of nonexpansive mapping in CAT(0) spaces. Open J. Math. Anal., 2017. Vol. 1. P. 1–12. DOI: 10.30538/psrp-oma2017.0001
  3. Agarwal R. P., O’Regan D., Sahu D. R. Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal., 2007. Vol. 8, No. 1. P. 61–79.
  4. Ambrosio L., Gigli N., Savare G. Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser, 2008. 334 p. DOI: 10.1007/978-3-7643-8722-8
  5. Ariza-Ruiz D., Leu¸stean L., López-Acedo G. Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Amer. Math. Soc., 2014. Vol. 366. No. 8. P. 4299–4322. DOI: 10.1090/S0002-9947-2014-05968-0
  6. Bačák M. The proximal point algorithm in metric spaces. Israel J. Math., 2013. Vol. 194. P. 689–701. DOI: 10.1007/s11856-012-0091-3
  7. Berg I. D., Nikolaev I. G. Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata, 2008. Vol. 133. P. 195–218. DOI: 10.1007/s10711-008-9243-3
  8. Bonyah E., Ahmad M., Ahmad I. On the viscosity rule for common fixed points of two nonexpansive mappings in CAT(0) spaces. Open J. Math. Sci., 2018. Vol. 2. No. 1. P. 39–55. DOI: 10.30538/oms2018.0016
  9. Bridson M. R., Häfliger A. Metric Spaces of Nonpositive Curvature. Grundlehren Math. Wiss., vol. 319. Berlin, Heidelberg: Springer-Verlag, 1999. 643 p. DOI: 10.1007/978-3-662-12494-9
  10. Burago D., Burago Yu., Ivanov S. A Course in Metric Geometry. Grad. Stud. Math., vol. 33. Providence, RI: A.M.S., 2001. 415 p.
  11. Chang S.-S., Wang L., Joseph Lee H. W., Chan C. K., Yang L. Demiclosed principle and △-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces. Appl. Math. Comput., 2012. Vol. 219, No. 5. P. 2611–2617. DOI: 10.1016/j.amc.2012.08.095
  12. Chang S.-S., Yao J.-C., Wang L., Qin L. J. Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl., 2016. Art. no. 68. P. 1–11. DOI: 10.1186/s13663-016-0559-7
  13. 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. Art. no. 227. P. 1–13. DOI: 10.1186/s13663-015-0465-4
  14. Dehghan H., Rooin J. A Characterization of Metric Projection in CAT(0) spaces. 2012. 3 p. arXiv: 1311.4174 [math.FA]
  15. Dhompongsa S., Kirk W. A., Sims B. Fixed points of uniformly lipschitzian mappings. Nonlinear Anal., 2006. Vol. 65, No. 4. P. 762–772. DOI: 10.1016/
  16. Dhompongsa S., Panyanak B. On △-convergence theorems in CAT(0) spaces. Comput. Math. Appl., 2008. Vol. 56, No. 10. P. 2572–2579. DOI: 10.1016/j.camwa.2008.05.036
  17. Jost J. Convex functionals and generalized harmonic maps into spaces of non positive curvature. Comment. Math. Helv., 1995. Vol. 70. P. 659–673. DOI: 10.1007/BF02566027
  18. Güler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim., 1991. Vol. 29, No. 2. P. 403–419. DOI: 10.1137/0329022
  19. Kakavandi B. A. Weak topologies in complete CAT(0) metric spaces. Proc. Amer. Math. Soc., 2012. Vol. 141, No. 3. P. 1029–1039. URL:
  20. Kamimura S., Takahashi W. Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory, 2000. Vol. 106, No. 2. P. 226–240. DOI: 10.1006/jath.2000.3493
  21. Kang S. M., Haq A. U., Nazeer W., Ahmad I., Ahmad M. Explicit viscosity rule of nonexpansive mappings in CAT(0) spaces. J. Comput. Anal. Appl., 2019. Vol. 27, No. 6. P. 1034–1043.
  22. Kirk W. A. Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis, Malaga/Seville, 2002/2003, Álvares D.G., Acelo G.L., Caro R.V. (eds.), vol. 64. P. 195–225.
  23. Kirk W. A. Geodesic geometry and fixed point theory, II. In: Int. Conf. on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, Japan, 2004. P. 113–142.
  24. Kirk W. A., Panyanak B. A concept of convergence in geodesic spaces. Nonlinear Anal., 2008. Vol. 68, No. 12. P. 3689–3696. DOI: 10.1016/
  25. Maingé P. E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal., 2008. Vol. 16, No. 7–8. P. 899–912. DOI: 10.1007/s11228-008-0102-z
  26. Martinet B. Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Opér., 1970. Vol. 4, No. R3. P. 154–158. (in France)
  27. Mayer U. F. Gradient flows on nonpositively curved metric spaces and harmonic maps. Commun. Anal. Geom. 1998. Vol. 6, No. 2. P. 199–253.
  28. Rockafellar R. T. Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 1976. Vol. 14, No. 5. P. 877–898. DOI: /10.1137/0314056
  29. Suparatulatorn R., Cholamjiak P., Suantai S. On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces. Optim. Methods Softw., 2017. Vol. 32, No. 1. P. 182–192. DOI: 10.1080/10556788.2016.1219908
  30. Xu H. K. Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2), 2002. Vol. 66, No. 1. P. 240–256. DOI: 10.1112/S0024610702003332


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