Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

CONVERGENCE THEOREMS OF MIXED TYPE IMPLICIT ITERATION FOR NONLINEAR MAPPINGS IN CONVEX METRIC SPACES

  • Kyung Soo, Kim (Department of Mathematics Education, Kyungnam University)
  • Received : 2022.06.10
  • Accepted : 2022.07.28
  • Published : 2022.12.06
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we propose and study an implicit iteration process for a finite family of total asymptotically quasi-nonexpansive mappings and a finite family of asymptotically quasi-nonexpansive mappings in the intermediate sense in convex metric spaces and establish some strong convergence results. Also, we give some applications of our result in the setting of convex metric spaces. The results of this paper are generalizations, extensions and improvements of several corresponding results.

Keywords

Acknowledgement

The author would like to thank the referees for their valuable comments and suggestions which improved the presentation of this paper. This work was supported by Kyungnam University Foundation Grant, 2022.

References

  1. S.S. Abed and Z.M.M. Hasan, Common fixed point of a finite-step iteration algorithm under total asymptotically quasi-nonexpansive maps, Baghdad Sci. J., 16(3) (2019), 654-660. https://doi.org/10.21123/bsj.2019.16.3.0654
  2. Ya.I. Alber, C.E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2006 (2006), Article ID 10673. Doi:10.1155/FPTA/2006/10673.
  3. I. Beg, M. Abbas and J.K. Kim, Convergence theorems of the iterative schemes in convex metric spaces, Nonlinear Funct. Anal. Appl., 11(3) (2006), 421-436.
  4. S.S. Chang, J.K. Kim and D.S. Jin, Iterative sequences with errors for asymptotically quasi-nonexpansive type mappings in convex metric spaces, Archives Ineq. Appl., 2 (2004), 365-374.
  5. S.S. Chang, L. Yang and X.R. Wang, Strong convergence theorems for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comput., 217 (2010), 277-282. https://doi.org/10.1016/j.amc.2010.05.058
  6. K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35(1) (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  7. A.R. Khan and M.A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Comput. Math. Appl., 59 (2010), 2990-2995. https://doi.org/10.1016/j.camwa.2010.02.017
  8. A.R. Khan, A.A. Domlo and H. Fukhar-ud-din, Common fixed points of Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11. https://doi.org/10.1016/j.jmaa.2007.06.051
  9. S.H. Khan, I. Yildirim and M. Ozdemir, Convergence of an implicit algorithm for two families of nonexpansive mappings, Comput. Math. Appl., 59 (2010), 3084-3091. https://doi.org/10.1016/j.camwa.2010.02.029
  10. J.K. Kim, K.H. Kim and K.S. Kim, Three-step iterative sequences with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces, Proc. RIMS Kokyuroku, Kyoto Univ., 1365 (2004), 156-165.
  11. J.K. Kim, K.S. Kim and S.M. Kim, Convergence theorems of implicit iteration process for finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Nonlinear Anal. Convex Anal., 1484 (2006), 40-51.
  12. K.S. Kim, Convergence of double acting iterative scheme for a family of generalized ϕ-weak contraction mappings in CAT(0) spaces, J. Comput. Anal. Appl., 27(3) (2019), 404-416.
  13. K.S. Kim, Convergence theorems for mixed type total asymptotically nonexpansive mappings in convex metric space, J. Nonlinear Conv. Anal., 21(9) (2020), 1931-1941.
  14. K.S. Kim, Convergence theorem for a generalized ϕ-weakly contractive nonself mapping in metrically convex metric spaces, Nonlinear Funct. Anal. Appl., 26(3) (2021), 601-610. https://doi.org/10.22771/NFAA.2021.26.03.10
  15. W.A. Kirk, Krasnoselskii's iteration process in hyperbolic space, Numer. Funct. Anal. Optim., 4 (1982), 371-381. https://doi.org/10.1080/01630568208816123
  16. Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl., 259 (2001), 18-24. https://doi.org/10.1006/jmaa.2000.7353
  17. G.S. Saluja, Convergence of fixed point of asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. Nonlinear Sci. Appl., 1 (2008), 132-144. https://doi.org/10.22436/jnsa.001.03.02
  18. G.S. Saluja, Approximating common fixed points for asymptotically quasi-nonexpansive mappings in the intermediate sense in convex metric spaces, Funct. Anal. Appro. Comput., 3(1) (2011), 33-44.
  19. G.S. Saluja, Convergence to common fixed points of multi-step iteration process for generalized asymptotically quasi-nonexpansive mappings in convex metric spaces, Hacettepe J. Math. Stat., 43(2) (2014), 205-221.
  20. G.S. Saluja and H.G. Hyun, An implicit iteration process for mixed type nonlinear mappings in convex metric spaces, Nonlinear Func. Anal. Appl., 22(4) (2017), 825-839. https://doi.org/10.22771/NFAA.2017.22.04.09
  21. G.S. Saluja and H.K. Nashine, Convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, Opuscula Math., 30(3) (2010), 331-340.
  22. Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351-358. https://doi.org/10.1016/S0022-247X(03)00537-7
  23. W. Takahashi, A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep., 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111
  24. R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58 (1992), 486-491. https://doi.org/10.1007/BF01190119
  25. H.K. Xu and R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), 767-773. https://doi.org/10.1081/NFA-100105317
  26. I. Yildirim and S.H. Khan, Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces, Appl. Math. Comput., 218 (2012), 4860-4866. https://doi.org/10.1016/j.amc.2011.10.049