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Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups

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

Abstract

In this paper, we introduce a new hybrid iterative process for finding a common element of the set of common fixed points of a finite family of nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove strong convergence of the proposed iterative process. The results we obtain extend and improve some recent known results.

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References

  1. Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Butnariu D., Censor Y., Gurfil P., Hadar E.: On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces. SIAM J. Optim. 19, 786–807 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hale E.T., Yin W., Zhang Y.: Fixed-point continuation applied to compressed sensing: implementation and numerical experiments. J. Comput. Math. 28, 170–194 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Maruster S., Popirlan C.: On the Mann-type iteration and the convex feasibility problem. J. Comput. Appl. Math. 212, 390–396 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byrne C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Censor Y., Elfving T., Kopf N., Bortfeld T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Prob. 21, 2071–2084 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xu H.K.: A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem. Inverse Prob. 22, 2021–2034 (2006)

    Article  MATH  Google Scholar 

  8. Podilchuk C.I., Mammone R.J.: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. A 7, 517–521 (1990)

    Article  Google Scholar 

  9. Youla, D.: Mathematical theory of image restoration by the method of convex projections. In: Star, H. (ed.) Image Recovery: Theory and Application, pp. 29–77. Academic Press, Orlando (1987)

  10. Browder F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1956)

    Article  MathSciNet  Google Scholar 

  11. Mann W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)

    Article  MATH  Google Scholar 

  12. Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Takahashi W., Takeuchi Y., Kubota R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Suzuki T.: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 131, 2133–2136 (2003)

    Article  MATH  Google Scholar 

  15. He, H., Chen, R.: Strong convergence theorems of the CQ method for nonexpansive semigroups. In: Fixed Point Theory and Applications. Article ID 59735, p 8 (2007)

  16. Saejung, S.: strong convergence theorems for nonexpansive semigroups without Bochner integrals. In: Fixed Point Theory and Applications. Article ID 745010, p 7 (2008)

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

    MathSciNet  MATH  Google Scholar 

  18. Flam S.D., Antipin A.S.: Equilibrium programming using proximal-link algolithms. Math. Program 78, 29–41 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moudafi, A., Thera, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Note in Economics and Mathematical Systems, vol. 477, pp 187–201. Springer, New York (1999)

  20. Tada A., Takahashi W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marino G., Xu H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space. J. Math. Anal. Appl. 329, 336–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Young Researchers Club, Babol Branch, Islamic Azad University, Babol, Iran

    Mohammad Eslamian

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  1. Mohammad Eslamian
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Correspondence to Mohammad Eslamian.

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Eslamian, M. Hybrid method for equilibrium problems and fixed point problems of finite families of nonexpansive semigroups. RACSAM 107, 299–307 (2013). https://doi.org/10.1007/s13398-012-0069-3

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  • DOI: https://doi.org/10.1007/s13398-012-0069-3

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