Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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AN ITERATIVE SCHEME FOR EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS OF ASYMPTOTICALLY k-STRICT PSEUDO-CONTRACTIVE MAPPINGS

  • Wang, Ziming (Department of Mathematics, Tianjin Polytechnic University) ;
  • Su, Yongfu (Department of Mathematics, Tianjin Polytechnic University)
  • Published : 2010.01.31
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Abstract

In this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an asymptotically k-strict pseudo-contractive mapping in the setting of real Hilbert spaces. We establish some weak and strong convergence theorems of the sequences generated by our proposed scheme. Our results are more general than the known results which are given by many authors. In particular, necessary and sufficient conditions for strong convergence of our iterative scheme are obtained.

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References

  1. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123–145.
  2. 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. (2008); doi:10.1016/j.cam.2008.03.032.
  3. L.-C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. (2007); doi:10.1016/j.cam.2007.02.022.
  4. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117–136.
  5. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program. 78 (1997), 29–41. https://doi.org/10.1016/S0025-5610(96)00071-8
  6. F. Flores-Bazan, Existence theory for finite-dimensional pseudomonotone equilibrium problems, Acta Appl. Math. 77 (2003), 249–297. https://doi.org/10.1023/A:1024971128483
  7. K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.
  8. N. Hadjisavvas, S. Komlsi, and S. Schaible, Handbook of Generalized Convexity and Generalized Monotonicity, Springer-Verlag, Berlin, Heidelberg, New York, 2005.
  9. N. Hadjisavvas and S. Schaible, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl. 96 (1998), 297–309. https://doi.org/10.1023/A:1022666014055
  10. T. H. Kim and H. K. Xu, Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions, Nonlinear Analysis-Theory Methods & Applications 68 (2008), 2828–2836. https://doi.org/10.1016/j.na.2007.02.029
  11. G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336–346. https://doi.org/10.1016/j.jmaa.2006.06.055
  12. A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl. 241 (2000), 46–55. https://doi.org/10.1006/jmaa.1999.6615
  13. M. O. Osilike and Y. Shehu, Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder-Petryshyn type, Nonlinear Analysis (2008); doi:10.1016/j.na.2008.07.015.
  14. A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama, 2006, pp. 609–617.
  15. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506–515. https://doi.org/10.1016/j.jmaa.2006.08.036