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Hybrid Algorithms for Ky Fan Inequalities and Common Fixed Points of Demicontractive Single-valued and Quasi-nonexpansive Multi-valued Mappings

  • Onjai-uea, Nawitcha (Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University) ;
  • Phuengrattana, Withun (Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University)
  • 투고 : 2017.12.30
  • 심사 : 2018.12.06
  • 발행 : 2019.12.23

초록

In this paper, we consider a common solution of three problems in real Hilbert spaces: the Ky Fan inequality problem, the variational inequality problem and the fixed point problem for demicontractive single-valued and quasi-nonexpansive multi-valued mappings. To find the solution we present a new iterative algorithm and prove a strong convergence theorem under mild conditions. Moreover, we provide a numerical example to illustrate the convergence behavior of the proposed iterative method.

키워드

참고문헌

  1. P. N. Anh, An LQ regularization method for pseudomonotone equilibrium problems on polyhedra, Vietnam J. Math. 36(2008), 209-228.
  2. P. N. Anh, Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities, J. Optim. Theory Appl., 154(2012), 303-320. https://doi.org/10.1007/s10957-012-0005-x
  3. P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62(2013), 271-283. https://doi.org/10.1080/02331934.2011.607497
  4. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63(1994), 123-145.
  5. H. Brezis, L. Nirenberg and G. Stampacchia, A remark on Ky Fan's minimax principle, Boll. Unione Mat. Ital., 6(1972), 293-300.
  6. F. Facchinei and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer-Verlag, New York, 2003.
  7. K. Fan, A minimax inequality and applications, Inequality III, pp. 103-113, Academic Press, New York, 1972.
  8. J. Garcia-Falset, E. Lorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl., 375(2011), 185-195. https://doi.org/10.1016/j.jmaa.2010.08.069
  9. F. Giannessi and A. Maugeri, Variational inequalities and network equilibrium problems, Plenum Press, New York, 1995.
  10. F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium problems: Nonsmooth optimization and variational inequality models, Kluwer Academic Publ., Dordrecht, 2004.
  11. K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, New York, 1984.
  12. T. L. Hicks and J. R. Kubicek, On the Mann iteration process in Hilbert spaces, J. Math. Anal. Appl., 59(1977), 498-504. https://doi.org/10.1016/0022-247X(77)90076-2
  13. H. Iiduka and W. Takahashi, Weak convergence theorems by Cesaro means for non-expansive mappings and inverse-strongly monotone mappings, J. Nonlinear Convex Anal., 7(2006), 105-113.
  14. F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91(2008), 166-177. https://doi.org/10.1007/s00013-008-2545-8
  15. G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekonom. i Mat. Metody, 12(1976), 747-756.
  16. P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16(2008), 899-912. https://doi.org/10.1007/s11228-008-0102-z
  17. L. D. Muu, V. H. Nguyen and T. D. Quoc, Extragradient algorithms extended to equilibrium problems, Optimization, 57(2008), 749-776. https://doi.org/10.1080/02331930601122876
  18. L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal., 18(1992), 1159-1166. https://doi.org/10.1016/0362-546X(92)90159-C
  19. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73(1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  20. 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
  21. S. Suantai, P. Cholamjiak, Y. J. Cho and W. Cholamjiak, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory Appl., (2016), Paper No. 35, 16 pp.
  22. S. Suantai and W. Phuengrattana, A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems, Banach J. Math. Anal., 11(2017), 661-675. https://doi.org/10.1215/17358787-2017-0010
  23. W. Takahashi, Nonlinear functional analysis, Yokohama Publishers, Yokohama, 2000.
  24. J. Vahidi, A. Latif and M. Eslamian, New iterative scheme with strict pseudocontractions and multivalued nonexpansive mappings for fixed point problems and variational inequality problems, Fixed Point Theory Appl., 2013, 2013:213, 13 pp.
  25. H. K. Xu, On weakly nonexpansive and ${\ast}$-nonexpansive multivalued mappings, Math. Japon., 36(1991), 441-445.
  26. H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116(2003), 659-678. https://doi.org/10.1023/A:1023073621589
  27. I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Stud. Comput. Math., 8(2001), 473-504. https://doi.org/10.1016/S1570-579X(01)80028-8
  28. H. Zegeye and N. Shahzad, Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62(2011), 4007-4014. https://doi.org/10.1016/j.camwa.2011.09.018