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A block hybrid method for solving generalized equilibrium problems and convex feasibility problem

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Abstract

A block hybrid projection algorithm for solving the convex feasibility problem and the generalized equilibrium problems for an infinite family of total quasi-ϕ-asymptotically nonexpansive mappings is introduced. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

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References

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

    Chapter  Google Scholar 

  2. Kitahara, S., Takahashi, W.: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 2, 333–342 (1993)

    MathSciNet  MATH  Google Scholar 

  3. O’Hara, J.G., Pillay, P., Xu, H.K.: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal. 64(9), 2022–2042 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang, S.S., Yao, J.C., Kim, J.K., Yang, L.: Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory and Applications, Article ID 46797, 19 p. (2008)

  5. Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 134, 257–266 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhou, H.Y., Gao, G., Tan, B.: Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings. J. Appl. Math. Comput. (2009). doi:10.1007/s12190-009-0263-4

    Google Scholar 

  8. Wattanawitoon, K., Kumam, P.: Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings. Nonlinear Anal., Hybrid Systems 3, 11–20 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, H.Y., Su, Y.F.: Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems. Nonlinear Anal. 72(2), 847–855 (2010)

    MathSciNet  Google Scholar 

  10. Takahashi, W., Zembayashi, K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kikkawa, M., Takahashi, W.: Approximating fixed points of nonexpansive mappings by the block iterative method in Banach spaces. Int. J. Comput. Numer., Anal. Appl. 5(1), 59–66 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Aleyner, A., Reich, S.: Block-iterative algorithms for solving convex feasibilitys in Hilbert and in Banach spaces. J. Math. Anal. Appl. 343(1), 427–435 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications Article ID 583082, 19 p. (2008) doi:10.1155/2008/583082

  14. Plubtieng, S., Ungchittrakool, K.: Strong convergence theorems of block iterative methods for a fonite family of relatively nonexpansive mappings in Banach spaces. J. Nonlinear Convex Anal. 8(3), 431–450 (2009)

    MathSciNet  Google Scholar 

  15. Sahu, D.R., Xu, H.K., Yao, J.C.: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. TMA 79(10), 3502–3511 (2009)

    Article  MathSciNet  Google Scholar 

  16. Chang, S.S., Joseph Lee, H.W., Chan, C.K.: A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications. Nonlinear Anal. TMA, 73, 2260–2270 (2010)

    Article  MATH  Google Scholar 

  17. Chang, S.S.: On the generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. 30(9), 1105–1112 (2009)

    Article  MathSciNet  Google Scholar 

  18. Chang, S.S., Kim, J.K., Wang, X.R.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. Article ID 869684, 14 p. (2010) doi:10.1155/2010/869684

  19. Kang, J., Su, Y., Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. HS, 4(2), 847–855 (2010)

    Google Scholar 

  20. Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990)

    Book  MATH  Google Scholar 

  21. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1/4), 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartosator, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Marcel Dekker, New York (1996)

    Google Scholar 

  23. Su, Y.F., Xu, H.K., Zhang, X.: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 73, 3890–3906 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chang, S.S., Joseph Lee, H.W., Chan, C.K., Yang, L.: Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications. Applied Math. Cmput. 218, 2921–2931 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, 650221, China

    Shih-sen Chang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hongkong, Hongkong

    H. W. Joseph Lee & Chi Kin Chan

Authors
  1. Shih-sen Chang
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  2. H. W. Joseph Lee
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  3. Chi Kin Chan
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Correspondence to Shih-sen Chang.

Additional information

Communicated by Lixin Shen.

This work was supported by the Natural Science Foundation of Yunnan University of Fanance and Economics.

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Chang, Ss., Lee, H.W.J. & Chan, C.K. A block hybrid method for solving generalized equilibrium problems and convex feasibility problem. Adv Comput Math 38, 563–580 (2013). https://doi.org/10.1007/s10444-011-9249-5

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  • DOI: https://doi.org/10.1007/s10444-011-9249-5

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