Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WEAK CONVERGENCE THEOREMS FOR 2-GENERALIZED HYBRID MAPPINGS AND EQUILIBRIUM PROBLEMS

  • Alizadeh, Sattar (Department of Mathematics Marand Branch, Islamic Azad University) ;
  • Moradlou, Fridoun (Department of Mathematics Sahand University of Technology)
  • Received : 2015.12.05
  • Published : 2016.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we propose a new modied Ishikawa iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of 2-generalized hybrid mappings in a Hilbert space. Our results generalize and improve some existing results in the literature. A numerical example is given to illustrate the usability of our results.

Keywords

References

  1. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer 2009.
  2. S. Alizadeh and F. Moradlou, Weak and strong convergence theorems for m-generalized hybrid mappings in Hilbert spaces, Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 315-328. https://doi.org/10.12775/TMNA.2015.049
  3. S. Alizadeh and F. Moradlou, A strong convergence theorem for equilibrium problems and generalized hybrid mappings, Mediterr. J. Math. 13 (2016), no. 1, 379-390. https://doi.org/10.1007/s00009-014-0462-6
  4. E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
  5. 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 (2009), no. 2, 967-974. https://doi.org/10.1016/j.cam.2008.03.032
  6. 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), no. 1, 186-201. https://doi.org/10.1016/j.cam.2007.02.022
  7. C. E. Chidume and S. A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2359-2363. https://doi.org/10.1090/S0002-9939-01-06009-9
  8. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.
  9. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997), no. 1, Ser. A, 29-41. https://doi.org/10.1016/S0025-5610(96)00071-8
  10. A. Genel and J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22 (1975), no. 1, 81-86. https://doi.org/10.1007/BF02757276
  11. F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academics Publishers, Dordrecht, Holland, 2001.
  12. M. Hojo, W. Takahashi, and I. Termwuttipong, Strong convergence theorem for 2-generalized hybrid mapping in Hilbert spaces, Nonlinear Anal. 75 (2012), no. 4, 2166-2176. https://doi.org/10.1016/j.na.2011.10.017
  13. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 40 (1974), 147-150.
  14. S. Itoh and W. Takahashi, The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. Math. 79 (1978), no. 2, 493-508. https://doi.org/10.2140/pjm.1978.79.493
  15. C. Jaiboon and P. Kumam, A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Fixed Point Theory Appl. 2009 (2009), Article ID 374815, 32 pages.
  16. C. Jaiboon and P. Kumam, and H. W. Humphries, Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems, Bull. Malays. Math. Sci. Soc. (2) 32 (2009), no. 2, 173-185.
  17. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
  18. F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), no. 2, 166-177. https://doi.org/10.1007/s00013-008-2545-8
  19. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  20. T. Maruyama, W. Takahashi, and M. Yao, Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal. 12 (2011), no. 1, 185-197.
  21. A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, Ill-posed variational problems and regularization techniques (Trier, 1998), 187-201, Lecture Notes in Econom. and Math. Systems, 477, Springer, Berlin, 1999.
  22. S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009), no. 2, 614-621. https://doi.org/10.1016/j.cam.2008.05.045
  23. 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), no. 1, 20-30. https://doi.org/10.1016/j.cam.2008.06.011
  24. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no. 1, 506-515. https://doi.org/10.1016/j.jmaa.2006.08.036
  25. W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohoma Publishers, Yokohoma, 2009.
  26. W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), no. 1, 79-88.
  27. W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428. https://doi.org/10.1023/A:1025407607560
  28. W. Takahashi and K. Zembayashi, Strong convergence theorems by hybrid methods for equilibrium problems and relatively nonexpansive mapping, Fixed Point Theory Appl. 2008 (2008), Article ID 528476, 11 pages.
  29. F. Yan, Y. Su, and Q. Feng, Convergence of Ishikawa method for generalized hybrid mappings, Commun. Korean Math. Soc. 28 (2013), no. 1, 135-141. https://doi.org/10.4134/CKMS.2013.28.1.135

Cited by

  1. Extragradient Methods for Solving Equilibrium Problems, Variational Inequalities, and Fixed Point Problems vol.38, pp.11, 2017, https://doi.org/10.1080/01630563.2017.1321017