Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERAL ITERATIVE ALGORITHMS FOR MONOTONE INCLUSION, VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS

  • Received : 2018.11.28
  • Accepted : 2021.03.10
  • Published : 2021.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce two general iterative algorithms (one implicit algorithm and one explicit algorithm) for finding a common element of the solution set of the variational inequality problems for a continuous monotone mapping, the zero point set of a set-valued maximal monotone operator, and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed iterative algorithms to a common point of three sets, which is a solution of a certain variational inequality. Further, we find the minimum-norm element in common set of three sets.

Keywords

Acknowledgement

The author would like to thank the anonymous reviewers for their careful reading and valuable comments along with providing a recent related paper, which improved the presentation of this manuscript.

References

  1. K. Afassinou, O. K. Narain, and O. E. Otunuga, Iterative algorithm for approximating solutions of split monotone variational inclusion, variational inequality and fixed point problems in real Hilbert spaces, Nonlinear Funct. Anal. Appl. 25 (2020) no.3, 491-510. https://doi.org/10.22771/NFAA.2020.25.03.06
  2. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, 6, Springer, New York, 2009. https://doi.org/10.1007/978-0-387-75818-3
  3. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. https://doi.org/10.1016/0022-247X(67)90085-6
  4. T. Chamnarnpan and P. Kumam, A new iterative method for a common solution of fixed points for pseudo-contractive mappings and variational inequalities, Fixed Point Theory Appl. 2012 (2012), 67, 15 pp. https://doi.org/10.1186/1687-1812-2012-67
  5. H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), no. 3, 341-350. https://doi.org/10.1016/j.na.2003.07.023
  6. J. S. Jung, A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces, J. Inequal. Appl. 2010 (2010), Art. ID 251761, 16 pp. https://doi.org/10.1155/2010/251761
  7. J. S. Jung, Iterative algorithms for monotone inclusion problems, fixed point problems and minimization problems, Fixed Point Theory Appl. 2013 (2013), 272, 23 pp. https://doi.org/10.1186/1687-1812-2013-272
  8. J. S. Jung, Strong convergence theorems for maximal monotone operators and continuous pseudocontractive mappings, J. Nonlinear Sci. Appl. 9 (2016), no. 6, 4409-4426. https://doi.org/10.22436/jnsa.009.06.81
  9. B. Martinet, Regularisation d'inequations variationnelles par approximations successives, Rev. Fran,caise Informat. Recherche Operationnelle 4 (1970), S'er. R-3, 154-158.
  10. G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc. 73 (1967), 315-321. https://doi.org/10.1090/S0002-9904-1967-11732-4
  11. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877-898. https://doi.org/10.1137/0314056
  12. N. Shahzad and H. Zegeye, Approximating a common point of fixed points of a pseudocontractive mapping and zeros of sum of monotone mappings, Fixed Point Theory Appl. 2014 (2014), 85, 15 pp. https://doi.org/10.1186/1687-1812-2014-85
  13. G. Stampacchia, Formes bilin'eaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413-4416.
  14. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), no. 1, 227-239. https://doi.org/10.1016/j.jmaa.2004.11.017
  15. S. Takahashi, W. Takahashi, and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl. 147 (2010), no. 1, 27-41. https://doi.org/10.1007/s10957-010-9713-2
  16. R. Wangkeeree and K. Nammanee, New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities, Fixed Point Theory Appl. 2013 (2013), 233, 17 pp. https://doi.org/10.1186/1687-1812-2013-233
  17. H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240-256. https://doi.org/10.1112/S0024610702003332
  18. I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, in Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000), 473-504, Stud. Comput. Math., 8, North-Holland, Amsterdam, 2001. https://doi.org/10.1016/S1570-579X(01)80028-8
  19. H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive mappings, ISRN Math. Anal. 2011 (2011), Art. ID 621901, 14 pp. https://doi.org/10.5402/2011/621901
  20. H. Zegeye and N. Shahzad, Strong convergence of an iterative method for pseudocontractive and monotone mappings, J. Global Optim. 54 (2012), no. 1, 173-184. https://doi.org/10.1007/s10898-011-9755-5