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A new hybrid projection algorithm for equilibrium problems and asymptotically quasi \(\phi \)-nonexpansive mappings in Banach spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we propose a new hybrid projection algorithm for finding a common element of the solution set of equilibrium problems and the fixed point set of asymptotically quasi \(\phi \)-nonexpansive mappings in strictly convex and uniformly smooth Banach spaces. A numerical example is given to illustrate the convergence of the proposed hybrid projection algorithms.

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References

  1. 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. Lecture Notes in Pure and Applied Mathematics, 15–50, vol. 17, pp. 15–50. Dekker, New York (1996)

    Google Scholar 

  2. Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings. J. Appl. Math. Comput. 48(1), 241–263 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. USA 54, 1041–1044 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cho, S.Y.: Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space. J. Appl. Anal. Comput. 8, 19–31 (2018)

    MathSciNet  Google Scholar 

  7. Chidume, C.E., Adamu, A., Okereke, L.C.: A Krasnoselskii-type algorithm for approximating solutions of variational inequality problems and convex feasibility problems. J. Nonlinear Var. Anal. 2, 203–218 (2018)

    Article  MATH  Google Scholar 

  8. Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. Mathematics and Its Applications, vol. 62. Kluwer, Dordrecht (1990)

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

    MathSciNet  MATH  Google Scholar 

  10. Hao, Y.: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl. 2013(227), 1–14 (2013)

    MathSciNet  Google Scholar 

  11. Hieu, D.V.: New subgradient extragradient methods for common solutions to equilibrium problems. Comput. Optim. Appl. 67(3), 571–594 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algor. 73(1), 197–217 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hieu, N.T., Dung, N.V.: Hybrid projection algorithm for two finite families of asymptotically quasi \(\phi \)-nonexpansive mappings in reflexive Banach spaces. Numer. Funct. Anal. Optim. 39(1), 67–86 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kazmi, K.R., Ali, R.: Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi-\(\phi \)-nonexpansive mapping in intermediate sense. Rev. R. Acad. Cienc. Exactas Fi’s. Nat. Ser. A Mat. RACSAM 111(3), 877 – 899 (2017)

  15. Lv, S.: Monotone projection methods for fixed points of asymptotically quasi-\(\phi \)-nonexpansive mappings. J. Nonlinear Funct. Anal. 2015, 1–13 (2015)

    Google Scholar 

  16. Qin, X., Cho, S.Y.: Convergence analysis of a monotone projection algorithm in reflexive Banach spaces. Acta Math. Sci. 37, 488–502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Qin, X., Cho, S.Y., Kang, S.M.: Strong convergence of shrinking projection methods for quasi \(\phi \)-nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 234, 750–760 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qin, X., Cho, S.Y., Kang, S.M.: On hybrid projection methods for asymptotically quasi-\(\phi \)-nonexpansive mappings. Appl. Math. Comput. 215, 3874–3883 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Qin, X., Cho, Y.J., Kang, S.M., Zhou, H.: Convergence of a modified Halpern-type iteration algorithm for quasi-\(\phi \)-nonexpansive mappings. Appl. Math. Lett. 22, 1051–1055 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Thakur, B.S., Thakur, D., Postolache, M.: A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. Appl. Math. Comput. 275, 147–155 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors sincerely thank two anonymous referees for several helpful comments. The authors also thank members of The Dong Thap Group of Mathematical Analysis and its Applications for their discussions on the manuscript.

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Authors and Affiliations

  1. Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nguyen Van Dung

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nguyen Van Dung

  3. Faculty of Mathematics Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap Province, Vietnam

    Nguyen Trung Hieu

Authors
  1. Nguyen Van Dung
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  2. Nguyen Trung Hieu
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Correspondence to Nguyen Trung Hieu.

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Van Dung, N., Hieu, N.T. A new hybrid projection algorithm for equilibrium problems and asymptotically quasi \(\phi \)-nonexpansive mappings in Banach spaces. RACSAM 113, 2017–2035 (2019). https://doi.org/10.1007/s13398-018-0595-8

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  • DOI: https://doi.org/10.1007/s13398-018-0595-8

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