WEAK CONVERGENCE OF AN ITERATIVE METHOD FOR EQUILIBRIUM PROBLEMS AND RELATIVELY NONEXPANSIVE MAPPINGS

  • Cho, Sun Young (Department of Mathematics Gyeongsang National University) ;
  • Kang, Shin Min (Department of Mathematics and the RINS Gyeongsang National University)
  • 투고 : 2009.08.24
  • 심사 : 2010.04.12
  • 발행 : 2010.06.30

초록

The purpose of this paper is to consider an iterative method for an equilibrium problem and a family relatively nonexpansive mappings. Weak convergence theorems are established in uniformly smooth and uniformly convex Banach spaces.

키워드

참고문헌

  1. Ya. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4(1994), 39-54.
  2. Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (Marcel Dekker, New York, 1996).
  3. D. Butnariu, S. Reich and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7(2001), 151-174. https://doi.org/10.1515/JAA.2001.151
  4. D. Butnariu, S. Reich and A. J. Zaslavski, Weak convergence of orbits of nonlin- ear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim. 24(2003), 489-508. https://doi.org/10.1081/NFA-120023869
  5. E. Blum and W. Oettli, From optimization and variational inequalities to equi- librium problems, Math. Student 63(1994), 123-145.
  6. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (Kluwer, Dordrecht, 1990).
  7. Y. J. Cho, S. M. Kang and X. Qin, Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Comput. Math. Appl. 56(2008), 2058-2064. https://doi.org/10.1016/j.camwa.2008.03.035
  8. V. Colao, G. L. Acedo and G. Marino, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal. 71(2009), 2708-2715. https://doi.org/10.1016/j.na.2009.01.115
  9. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6(2005), 117-136.
  10. O. Chadli, N. C. Wong and J. C. Yao, Equilibrium problems with applications to eigenvalue problems, J. Optim. Theory Appl. 117(2003), 245-266. https://doi.org/10.1023/A:1023627606067
  11. S. S. Chang, H. W. J. Lee and C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. 70(2009) 3307-3319. https://doi.org/10.1016/j.na.2008.04.035
  12. S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algo- rithm in a Banach space, SIAM J. Optim. 13(2002), 938-945 https://doi.org/10.1137/S105262340139611X
  13. S. Y. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2004(2004), 37-47. https://doi.org/10.1155/S1687182004310089
  14. S. Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134(2005), 257- 266. https://doi.org/10.1016/j.jat.2005.02.007
  15. A. Moudafi, Weak convergence theorems nonexpansive mappings and for equi- librium problems, J. Nonlinear Convex Anal. 9(2008), 37-43.
  16. X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225(2009), 20-30. https://doi.org/10.1016/j.cam.2008.06.011
  17. X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling. 48(2008), 1033-1046. https://doi.org/10.1016/j.mcm.2007.12.008
  18. X. Qin, Y. J. Cho and J. I. Kang, S. M. Kang, Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, J. Comput. Appl. Math. 230(2009), 121-127. https://doi.org/10.1016/j.cam.2008.10.058
  19. S. Reich, A weak convergence theorem for the alternating method with Breg- man distance, in: A.G. Kartsatos (Ed.), Theory and Applicationsof Nonlinear Operators of Accretive and Monotone Type (Marcel Dekker, New York, 1996).
  20. W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70(2009), 45-57. https://doi.org/10.1016/j.na.2007.11.031
  21. W. Takahashi, Nonlinear Functional Analysis (Yokohama-Publishers, 2000).
  22. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16(1991), 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K
  23. D. C. Youla, Mathematical theory of image restoration by the method of con- vex projections, in: H. Stark (Ed.), Image Recovery: Theory and Applications, Academic Press, Florida, 1987, 29-77.