DOI QR코드

DOI QR Code

The Three-step Intermixed Iteration for Two Finite Families of Nonlinear Mappings in a Hilbert Space

  • Suwannaut, Sarawut (Department of Mathematics, Faculty of Science, Lampang Rajabhat University) ;
  • Kangtunyakarn, Atid (Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang)
  • Received : 2019.06.16
  • Accepted : 2020.11.16
  • Published : 2022.03.31

Abstract

In this work, the three-step intermixed iteration for two finite families of nonlinear mappings is introduced. We prove a strong convergence theorem for approximating a common fixed point of a strict pseudo-contraction and strictly pseudononspreading mapping in a Hilbert space. Some additional results are obtained. Finally, a numerical example in a space of real numbers is also given and illustrated.

Keywords

Acknowledgement

This work was supported by the Research Administration Division of King Mongkut's Institute of Technology Ladkrabang and Lampang Rajabhat University.

References

  1. B. C. Deng, T. Chen and F. L. Li, Viscosity iteration algorithm for a ρ-strictly pseudononspreading mapping in a Hilbert space, J. Inequal. Appl., 80(2013).
  2. S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal. Theory Methods Appl., 71(2009), 2082-2089.
  3. S. Ishikawa, Fixed point by a new iterative method, Proc. Am. Math. Soc., 44(1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  4. A. Kangtunyakarn, Convergence theorem of κ-strictly pseudo-contractive mapping and a modification of genealized equilibrium problems, Fixed Point Theory Appl., 89(2012), 1-17.
  5. W. Khuangsatung and A. Kangtunyakarn, Algorithm of a new variational inclusion problem and strictly pseudononspreding mapping with application, Fixed Point Theory Appl., 209(2014), 22pp.
  6. F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91(2008), 166-177. https://doi.org/10.1007/s00013-008-2545-8
  7. Y. Kurokawa and W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. Theory Methods Appl., 73(2010), 1562-1568. https://doi.org/10.1016/j.na.2010.04.060
  8. H. Liu, J. Wang and Q. Feng, Strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space, Abstr. Appl. Anal. 2012, 11pp. https://doi.org/10.1155/AAA.2005.11
  9. W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4(1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  10. G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces J. Math. Anal. Appl., 318(2006), 43-52. https://doi.org/10.1016/j.jmaa.2005.05.028
  11. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241(2000), 46-55. https://doi.org/10.1006/jmaa.1999.6615
  12. Z. Opial, Weak convergence of the sequence of successive approximation of nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  13. M. O. Osilike and F. O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Anal., 74(2011), 1814-1822. https://doi.org/10.1016/j.na.2010.10.054
  14. S. Suwannaut and A. Kangtunyakarn, Convergence theorem for solving the combination of equilibrium problems and fixed point problems in Hilbert spaces, Thai J. Math., 14(2016), 77-79.
  15. S. Suwannaut, The S-intermixed iterative method for equilibrium problems, Thai J. Math., 17(2019), 60-74.
  16. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, (2000).
  17. H. K. Xu, An iterative approach to quadric optimization, J. Optim Theory Appl., 116(2003), 659-678. https://doi.org/10.1023/A:1023073621589
  18. Z. Yao, S. M. Kang and H. J. Li, An intermixed algorithm for strict pseudo-contractions in Hilbert spaces, Fixed Point Theory Appl., 206(2015), 11pp.