Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
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A GENERAL ITERATIVE ALGORITHM COMBINING VISCOSITY METHOD WITH PARALLEL METHOD FOR MIXED EQUILIBRIUM PROBLEMS FOR A FAMILY OF STRICT PSEUDO-CONTRACTIONS

  • Jitpeera, Thanyarat (Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi(KMUTT)) ;
  • Inchan, Issara (Department of Mathematics and Computer, Uttaradit Rajabhat University) ;
  • Kumam, Poom (Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi(KMUTT))
  • 투고 : 2010.07.08
  • 심사 : 2010.10.13
  • 발행 : 2011.05.30
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초록

The purpose of this paper is to introduce a general iterative process by viscosity approximation method with parallel method to ap-proximate a common element of the set of solutions of a mixed equilibrium problem and of the set of common fixed points of a finite family of $k_i$-strict pseudo-contractions in a Hilbert space. We obtain a strong convergence theorem of the proposed iterative method for a finite family of $k_i$-strict pseudo-contractions to the unique solution of variational inequality which is the optimality condition for a minimization problem under some mild conditions imposed on parameters. The results obtained in this paper improve and extend the corresponding results announced by Liu (2009), Plubtieng-Panpaeng (2007), Takahashi-Takahashi (2007), Peng et al. (2009) and some well-known results in the literature.

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연구 과제 주관 기관 : National Research Council of Thailand

참고문헌

  1. F. Flores-Bazan, Existence theorems for generalized noncoercive equilibrium problems: the quasiconvex case, SIAM Journal on Optimization, vol. 11, no. 3 (2000) 675-690, . No. https://doi.org/10.1137/S1052623499364134
  2. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium prob- lems, Math. Student. 63(1994) 123-145.
  3. F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Natl. Acad. Sci. USA 53 (1965) 1272-1276.
  4. F.E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear map- pings in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967) 82-90.
  5. F.E. Browder, 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
  6. L.C. Ceng, S. Al-Homidan, Q.H. Ansari and J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math., 223 (2) (2009), 967-974. https://doi.org/10.1016/j.cam.2008.03.032
  7. 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
  8. P. L. Combettes and S. A. Hirstoaga, Equilibrium programing using proximal-like algo- rithms, Math. Program. 78 (1997) 29-41.
  9. P. Kumam, A new hybrid iterative method for solution of Equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, Journal of Applied Mathematics and Computing, Volume 29, Issue 1 (2009) 263-280.
  10. P. Kumama and C. Jaiboon, A new hybrid iterative method for mixed equilibrium prob- lems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Analysis: Hybrid Systems 3 (2009) 510-530. https://doi.org/10.1016/j.nahs.2009.04.001
  11. P. Kumama and P. Katchang, A viscosity of extragradient approximation method for find- ing equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings, Nonlinear Analysis: Hybrid Systems 3 (2009) 475-486. https://doi.org/10.1016/j.nahs.2009.03.006
  12. W. Kumam and P. Kumam, Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization, Nonlinear Analysis: Hybrid Systems, 3 (2009) 640-656. https://doi.org/10.1016/j.nahs.2009.05.007
  13. P. Kumam, N. Petrot and R. Wangkeeree, A hybrid iterative scheme for equilibrium prob- lems and fixed point problems of asymptotically k-strictly pseudo-contractions, Journal of Computational and Applied Mathematics, 233 (2010) 2013-2026. https://doi.org/10.1016/j.cam.2009.09.036
  14. Y. Liu, A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., (in press) doi:10.1016/j.na.2009.03.060
  15. G. Marino and H. K. Xu, A general iterative method for nonexpansive mapping in Hilbert sapces, J. Math. Anal. Appl. 318 ( 2006) 43-52. https://doi.org/10.1016/j.jmaa.2005.05.028
  16. S. Plubtieng and R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math Anal. Appl. 336 (2007), 455-469. https://doi.org/10.1016/j.jmaa.2007.02.044
  17. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequence for one- parameter nonexpansive semigroups without Bochner integrals, J. Math Anal. Appl. 305 (2005) 227-239. https://doi.org/10.1016/j.jmaa.2004.11.017
  18. S. Takahashi and W. Takahashi, Viscosity approxmation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331(1) (2007) 506-515. https://doi.org/10.1016/j.jmaa.2006.08.036
  19. S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mappings in a Hilbert spaces, Nonlinear Anal. 69 (2008) 1025- 1033. https://doi.org/10.1016/j.na.2008.02.042
  20. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118 (2003) 417-428. https://doi.org/10.1023/A:1025407607560
  21. J. W. Peng, Iterative algorithms for mixed equilibrium problems, stric pseudocontractions and monotone mappings, J. Optim. Theory Appl, doi: 10.1007/s10957-009-9585-5
  22. J. W. Peng, Y. C. Liou and J. C. Yao, An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions, Fixed Point Theory and Applications Volume 2009, Article ID 794178, 21 pages.
  23. J. W. Peng and J. C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. and Comp. Model. 49 (2009) 1816-1828. https://doi.org/10.1016/j.mcm.2008.11.014
  24. J W. Peng and J. C. Yao, Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping, J Glob. Optim. doi: 10.1007/s10898-009-9428-9
  25. R. Wangkeeree and R. Wangkeeree, A general iterative method for variational inequality problems, mixed equilibrium problems and fixed point problems of strictly pseudocontrac- tive mappings in Hilbert spaces, Fixed Point Theory and Applications. Volum 2009 (2009), Article ID 519065, 32 pages.
  26. H. K. Xu, An iterative approach to guadratic optimazation, J. Optim. Theory Appl. 116 (2003) 659-678. https://doi.org/10.1023/A:1023073621589
  27. H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math Anal. Appl. 298 (2004), 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059
  28. Y. Yao, Y. C. Liou and R. Chen, Convergence theorem for for fixed point problems and variational inequality problems, Nonlinear Anal. 70 (5) (2009) 1956-1964. https://doi.org/10.1016/j.na.2008.02.094
  29. J. C. Yao and O.Chadli, Pseudomonotone complementarity problems and variational in- equalities,in: J.P. Crouzeix, N. Haddjissas, S. Schaible (Eds.),Handbook ofGeneralized Con- vexity and Monotonicity, Kluwer Academic, (2005) 501-558.
  30. H. Zhou, Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert space, Nonlinear Anal. , 69 (2) (2008) 456-462. https://doi.org/10.1016/j.na.2007.05.032