DOI QR코드

DOI QR Code

STRONG AND ∆-CONVERGENCE THEOREMS FOR A COUNTABLE FAMILY OF MULTI-VALUED DEMICONTRACTIVE MAPS IN HADAMARD SPACES

  • 투고 : 2021.02.14
  • 심사 : 2021.12.09
  • 발행 : 2022.03.15

초록

In this paper, iterative algorithms for approximating a common fixed point of a countable family of multi-valued demicontractive maps in the setting of Hadamard spaces are presented. Under different mild conditions, the sequences generated are shown to strongly convergent and ∆-convergent to a common fixed point of the considered family, accordingly. Our theorems complement many results in the literature.

키워드

참고문헌

  1. A. Abkar and M. Eslamian, Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces, Nonlinear Anal., 75 (2012), 1895- 1903. https://doi.org/10.1016/j.na.2011.09.040
  2. 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. https://doi.org/10.22771/NFAA.2021.26.05.14
  3. M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, 1999.
  4. F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl., 20 (1967), 197-229. https://doi.org/10.1016/0022-247x(67)90085-6
  5. P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983-987. https://doi.org/10.1016/j.jmaa.2005.08.006
  6. C.E. Chidume, A.U. Bello and P. Ndambomve, Strong and ∆-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr. Appl. Anal., 2014, 2014:6.
  7. C.E. Chidume, C.O. Chidume, N. Djitt'e and M.S. Minjibir, Convergence theorems for fixed points of multivalued strictly pseudocontractive mappings in Hilbert spaces, Abstr. Appl. Anal., 2013, Article ID 629468, 10 pages, 2013.
  8. C.E. Chidume and J.N. Ezeora, Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-conttractive mappings, Fixed Point Theory and Appl., 2014, article 111, 2014.
  9. S. Dhompongsa, A. Kaewkhao and B. Panyanak, On Kirk's strong convergence theorem for multivalued nonexpansive mappings on CAT (0) spaces, Nonlinear Anal., 75 (2012), 459-468. https://doi.org/10.1016/j.na.2011.08.046
  10. S. Dhompongsa and B. Panyanak, On ∆-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036
  11. 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
  12. S. Dhompongsa, W. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 35-45.
  13. G.Z. Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algor., 80 (2019), 1155-1179. https://doi.org/10.1007/s11075-018-0521-3
  14. G.Z. Eskandani, S. Azarmi and M. Raeisi, Products of resolvents and multivalued hybrid mappings in CAT(0) spaces, Acta Math. Sci., 38 (2018), 791-804. https://doi.org/10.1016/S0252-9602(18)30784-7
  15. T.L. Hicks and J.D. Kubicek, On the Mann Iteration Process in a Hilbert Space, J. Math. Anal. Appl., 59 (1977), 498-504. https://doi.org/10.1016/0022-247x(77)90076-2
  16. S.H. Khan and M. Abbas, Strong and ∆-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl., 61 (2011), 109-116. https://doi.org/10.1016/j.camwa.2010.10.037
  17. 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. https://doi.org/10.22771/NFAA.2018.23.01.07
  18. 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
  19. W.A. Kirk, Geodesic geometry and fixed point theory II, in: International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2004, pp. 113-142.
  20. W.A. Kirk, Krasnoselskii's iteration process in hyperbolic space, 4(4) (1981-1982), 371-381.
  21. U. Kohlenbach and L. Leustean, Mann iterates of directionally nonexpansive mappings in hyperbolic spaces, Abst. Appl. Anal., 2003:8 (2003), 449-477. https://doi.org/10.1155/S1085337503212021
  22. K. Lerkchaiyaphum and W. Phuengrattana, Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces, Numer. Algor., 77 (2018), 727-740. https://doi.org/10.1007/s11075-017-0337-6
  23. 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
  24. G.A. Okeke, M. Abbas and M. de la Sen, Fixed point theorems for convex minimization problems in CAT(0) spaces, Nonlinear Funct. Anal. Appl., 25(4) (2020), 671-696. https://doi.org/10.22771/NFAA.2020.25.04.04
  25. S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic space, Nonlinear Anal., 15 (1990), 537-558. https://doi.org/10.1016/0362-546X(90)90058-O
  26. S. Saejung, Halpern's Iteration in CAT(0) Spaces, Fixed Point Theory Appl., 2010, 471781 (2009).
  27. J. Tang, J. Zhu, S.S. Chang, M. Liu and X. Li, A new modified proximal point algorithm for a finite family of minimization problem and fixed point for a finite family of demicontractive mappings in Hadamard spaces, Nonlinear Funct. Anal. Appl., 25(3) (2020), 563-577. https://doi.org/10.22771/NFAA.2020.25.03.11