ESTONIAN ACADEMY
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eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
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proceedings
of the estonian academy of sciences
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Convergence theorems on generalized equilibrium problems and fixed point problems with applications; pp. 170–183

Full article in PDF format | doi: 10.3176/proc.2009.3.04

Authors
Xiaolong Qin, Shin Min Kang, Yeol Je Cho

Abstract
In this paper, we introduce an iterative algorithm for finding a common element in the set of solutions to generalized equilibrium problems and a set of fixed points of strict pseudo-contractions. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper mainly improve on the corresponding results reported by many others.
References

 1. Blum, E. and Oettli, W. From optimization and variational inequalities to equilibrium problems. Math. Stud., 1994, 63, 123–145.

  2. Browder, F. E. Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure. Math., 1985, 18, 78–81.

  3. Bruck, R. E. Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Amer. Math. Soc., 1973, 179, 251–262.
doi:10.2307/1996502

  4. Ceng, C. L. and Yao, J. C. Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput., 2008, 198, 729–741.
doi:10.1016/j.amc.2007.09.011

  5. Ceng, C. L. and Yao, J. C. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math., 2008, 214, 186–201.
doi:10.1016/j.cam.2007.02.022

  6. Censor, Y. and Zenios, S. A. Parallel Optimization: Theory, Algorithms, and Applications (Numerical Mathematics and Scientific Computation). Oxford University Press, New York, 1997.

  7. Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl., 2008, 344, 340–352.
doi:10.1016/j.jmaa.2008.02.041

  8. Combettes, P. L. The convex feasibility problem in image recovery. In Advances in Imaging and Electron Physics (Hawkes, P., ed.), vol. 95, pp. 155–270. Academic Press, New York, 1996.

  9. Combettes, P. L. and Hirstoaga, S. A. Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal., 2005, 6, 117–136.

10. Iiduka, H. and Takahashi, W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal., 2005, 61, 341–350.
doi:10.1016/j.na.2003.07.023

11. Kotzer, T., Cohen, N., and Shamir, J. Images to ration by a novel method of parallel projection onto constraint sets. Opt. Lett., 1995, 20, 1172–1174.
doi:10.1364/OL.20.001172

12. Moudafi, A. and Théra, M. Proximal and dynamical approaches to equilibrium problems. In Lecture Notes in Economics and Mathematical Systems, No. 477, pp. 187–201. Springer, 1999.

13. Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl., 2007, 336, 455–469.
doi:10.1016/j.jmaa.2007.02.044

14. Plubtieng, S. and Punpaeng, R. A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput., 2008, 197, 548–558.
doi:10.1016/j.amc.2007.07.075

15. Qin, X., Cho, Y. J., and Kang, S. M. Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math., 2009, 225, 20–30.
doi:10.1016/j.cam.2008.06.011

16. Qin, X., Shang, M., and Su, Y. Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model., 2008, 48, 1033–1046.
doi:10.1016/j.mcm.2007.12.008

17. Qin, X., Shang, M., and Su, Y. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal., 2008, 69, 3897–3909.
doi:10.1016/j.na.2007.10.025

18. Sezan, M. I. and Stark, I. Application of convex projection theory to image recovery in tomograph and related areas. In Image Recovery: Theory and Application (Stark, H., ed.), pp. 155–270. Academic Press, Orlando, 1987.

19. Shang, M., Su, Y., and Qin, X. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Fixed Point Theory Appl., 2007, 2007, Art. ID 95412.

20. Su, Y., Shang, M., and Qin, X. An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal., 2008, 69, 2709–2719.
doi:10.1016/j.na.2007.08.045

21. Suzuki, T. Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl., 2005, 305, 227–239.
doi:10.1016/j.jmaa.2004.11.017

22. Tada, A. and Takahashi, W. Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. J. Optim. Theory Appl., 2007, 133, 359–370.
doi:10.1007/s10957-007-9187-z

23. Takahashi, S. and Takahashi, W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl., 2007, 331, 506–515.
doi:10.1016/j.jmaa.2006.08.036

24. Takahashi, S. and Takahashi, W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal., 2008, 69, 1025–1033.
doi:10.1016/j.na.2008.02.042

25. Takahashi, W. and Zembayashi, K. Strong and weak convergence theorems for equilibrium problems and relatively non-expansive mappings in Banach spaces. Nonlinear Anal., 2009, 70, 45–57.
doi:10.1016/j.na.2007.11.031

26. Xu, H. K. Iterative algorithms for nonlinear operators. J. London Math. Soc., 2002, 66, 240–256.
doi:10.1112/S0024610702003332

27. Yao, Y., Noor, M. A., and Liou, Y. C. On iterative methods for equilibrium problems. Nonlinear Anal., 2009, 70, 497–509.
doi:10.1016/j.na.2007.12.021

28. Zhou, H. Convergence theorems of fixed points for k-strict seudo-contractions in Hilbert spaces. Nonlinear Anal., 2008, 69, 456–462.
doi:10.1016/j.na.2007.05.032


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