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CONVERGENCE THEOREMS FOR TWO NONLINEAR MAPPINGS IN CAT(0) SPACES

  • Sokhuma, Kritsana (Department of Mathematics, Faculty of Science and Technology, Phranakhon Rajabhat University) ;
  • Sokhuma, Kasinee (Department of Educational Research and Measurement, Faculty of Education, Nakhon Pathom Rajabhat University)
  • Received : 2021.02.26
  • Accepted : 2021.03.31
  • Published : 2022.09.01

Abstract

In this paper, we construct an iteration scheme involving a hybrid pair of the Suzuki generalized nonexpansive single-valued and multi-valued mappings in a complete CAT(0) space. In process, we remove a restricted condition (called end-point condition) in Akkasriworn and Sokhuma's results [2] in Banach spaces and utilize the same to prove some convergence theorems. The results in this paper, are analogs of the results of Akkasriworn et al. [3] in Banach spaces.

Keywords

Acknowledgement

The author thanks for the support of the Institute for Research and Development, Phranakhon Rajabhat University.

References

  1. A. Abkar and M. Eslamian, Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Space, Fixed Point Theory Appl., (2010); doi:10.1155/2010/457935.
  2. N. Akkasriworn and K. Sokhuma, Convergence theorem for a pair of asymptotically and multivalued nonexpansive mapping, Commun. Korean Math. Soc., 30 (2015), 177-189. https://doi.org/10.4134/CKMS.2015.30.3.177
  3. N. Akkasriworn, K. Sokhuma and K. Chuikamwong, Ishikawa iterative process for a pair of Suzuki generalized nonexpansive single valued and multivalued mappings in Banach spaces, Int. J. Math. Anal., 19 (2012), 923-932.
  4. M. Asadi, Sh. Ghasemzadehdibagi, S. Haghayeghi and N. Ahmad, Fixed point theorems for (α, p)-nonexpansive mappings in CAT(0) spaces, Nonlinear Funct. Anal. Appl., 26(5) (2021), 1045-1057.
  5. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
  6. F. Bruhat and J. Tits, Groupes reductifs sur un corps local I. Donnees radicielles valuees, Inst. Hautes Etudes Sci. Publ. Math., 41 (1972), 5-251. https://doi.org/10.1007/BF02715544
  7. S. Dhompongsa, W.A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35-45.
  8. S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), 762-772. https://doi.org/10.1016/j.na.2005.09.044
  9. R. Espinola, P. Lorenzo and A. Nicolae, Fixed points, selections and common fixed points for nonexpansive-type Mappings, J. Math. Anal. Appl., 382 (2011), 503-515. https://doi.org/10.1016/j.jmaa.2010.06.039
  10. J.K. Kim, R.P. Pathak, S. Dashputre, S.D. Diwan and R.L. Gupta, Demiclosedness principle and convergence theorems for Lipschitzian type nonself-mappings in CAT(0) spaces, Nonlinear Funct. Anal. Appl., 23(1) (2018), 73-95.
  11. K.S. Kim, Existence theorem of a fixed point for asymptotically nonexpansive nonself mapping in CAT(0) spaces, Nonlinear Funct. Anal. Appl., 25(2) (2020), 355-362.
  12. W.A. Kirk, Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Colecc. Abierta, vol.64, Univ. Sevilla Secr. Publ., Seville, 6 (2003), 195-225.
  13. W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696. https://doi.org/10.1016/j.na.2007.04.011
  14. T. Laokul and B. Panyanak, Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces, Int. J. Math. Anal., 3 (2009), 1305-1315.
  15. W. Laowang and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, Fixed Point Theory Appl., (2010); doi:10.1155/2010/367274.
  16. T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182. https://doi.org/10.1090/S0002-9939-1976-0423139-X
  17. B. Nanjaras, B. Panyanaka and W. Phuengrattana, Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces, Nonlinear Analysis: Hybrid Syst., 4 (2010), 25-31. https://doi.org/10.1016/j.nahs.2009.07.003
  18. G.A. Okeke, M. Abbas and M. de la Sen, Fixed point theorems for convex minimization problems in complex valued CAT(0) spaces, Nonlinear Funct. Anal. Appl., 25(4) (2020), 671-696.
  19. K. Sokhuma, ∆-Convergence Theorems for a Pair of Single valued and Multivalued Nonexpansive Mappings in CAT(0) spaces, J. Math. Anal., 4 (2013), 23-31.
  20. K. Sokhuma, An Ishikawa Iteration Scheme for two Nonlinear Mappings in CAT(0) Spaces, Kyungpook Math. J., 59 (2019), 665-678.
  21. K. Sokhuma and A. Kaewkhao, Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces, Fixed Point Theory Appl., (2010); doi:10.1155/2010/618767.
  22. T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mapping, J. Math. Anal. Appl., 340 (2008), 1088-1095. https://doi.org/10.1016/j.jmaa.2007.09.023