Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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A HYBRID METHOD FOR A SYSTEM INVOLVING EQUILIBRIUM PROBLEMS, VARIATIONAL INEQUALITIES AND NONEXPANSIVE SEMIGROUP

  • THUY, LE QUANG (School of Applied Mathematics and Informatics Hanoi University of Science and Technology) ;
  • MUU, LE DUNG (Institute of Mathematics Vietnam Academy of Science and Technology)
  • Received : 2014.12.24
  • Accepted : 2015.09.10
  • Published : 2015.09.30
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Abstract

In this paper we propose an iteration hybrid method for approximating a point in the intersection of the solution-sets of pseudomonotone equilibrium and variational inequality problems and the fixed points of a semigroup-nonexpensive mappings in Hilbert spaces. The method is a combination of projection, extragradient-Armijo algorithms and Manns method. We obtain a strong convergence for the sequences generated by the proposed method.

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References

  1. E.Blum and W.Oettli, From optimization and variational inequality to equilibrium problems, Math. Student 63 (1994), 127-149.
  2. F.E.Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. 54 (1965), 1041-1044. https://doi.org/10.1073/pnas.54.4.1041
  3. N.Buong, Strong convergence of a method for variational inequality problems and fixed point problems of a nonexpansive semigroup in Hilbert spaces, J. appl. math. inform. 20 (2011), 61-74.
  4. L.C.Ceng and J.C.Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186-201. https://doi.org/10.1016/j.cam.2007.02.022
  5. S.Chang, J.K.Kim and L.Wang, Total quasi--asymptotically nonexpansive semi-groups and strong convergence theorems in Banach spaces, Fixed Point Theory Appl. 1 (2012), 1-14.
  6. P.Daniele, F.Giannessi and A.Maugeri, Equilibrium problems and variational models, Kluwer, (2003).
  7. K.Goebel and W.A.Kirk, Topics in metric fixed point theory of cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, Mass, USA, (1990).
  8. U.Kamraksa and R.Wangkeeree, Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces, J. Glob. Optim. 51 (2011), 689-714. https://doi.org/10.1007/s10898-011-9654-9
  9. S.M.Kang, S.Y.Cho and Y.C.Kwun, Srong convergence of paths for nonexpansive semigroups in Banach space, Korean J. Math. 19 (2011), 279-289. https://doi.org/10.11568/kjm.2011.19.3.279
  10. J.K.Kim and N.Buong, A new explicit iteration method for variational inequali-ties on the set of common fixed points for a finite family of nonexpansive mappings, J. Inequal. Appl. (2013), Doi:10.1186/1029-242X-2013-419.
  11. J.K.Kim,Y.M.Nam and B.J.Jin, Weak convergence theorems for almost-orbits of an asymptotically nonexpansive semigroup in Banach spaces, Comm. Korean Math. Soc. 13 (1998), 501-513.
  12. N.Nadezhkina and W.Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241. https://doi.org/10.1137/050624315
  13. Z.Opial, Weak convergence of the sequence of successive approximations for non-expansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
  14. Q.Jiang and J.Wang, Hybrid algorithms of nonexpansive semigroup for mixed equilibrium problems, varitional inequalities and fixed point problems, J. Inequal. Appl. 174 (2014), Doi: 10.1186/1029-242X.
  15. R.T.Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88. https://doi.org/10.1090/S0002-9947-1970-0282272-5
  16. S.Saeidi, Iterative algorithms for finding common solutions of variational in-equalities and systems of equilibrium problems and fixed points of families and semi-groups of nonexpansive mappings, Nonlinear Anal. 70 (2009), 4195-4208. https://doi.org/10.1016/j.na.2008.09.009
  17. Y.Shehu, Aniterative method for nonexpansive semigroup, variational inclusions and generalized equilibrium problems, Math. Comput. Modelling 55 (2012), 1301-1314. https://doi.org/10.1016/j.mcm.2011.10.008
  18. T.Shimizu and W.Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), 71-83. https://doi.org/10.1006/jmaa.1997.5398
  19. T.Suzuki, Characterrizations of common fixed points of one-parameter nonexpansive semigroups, and convergence theorems to common fixed points, J. Math. Anal. Appl. 324 (2006), 1006-1019. https://doi.org/10.1016/j.jmaa.2006.01.004
  20. N.T.T. Thuy, Hybrid Mann-Halpern method for finding fixed point involving asymtotically nonexpansive mappings and semigroups Vietnam J. Math. 42 (2014), 219-232. https://doi.org/10.1007/s10013-014-0071-5
  21. P.T.Vuong, J.J.Strodiot and N.V.Hien, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl. 155 (2012), 605-627. https://doi.org/10.1007/s10957-012-0085-7
  22. P.Yang, Y.Yao, Y.C.Liou and R.Chen, Hybrid algorithms of nonexpansive semi-group for varitional inequalities, J. Appl. Math. Article ID 634927 (2012), Doi: 10.1155/2012/634927.

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