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A general iterative method of fixed points for equilibrium problems and optimization problems

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Abstract

The purpose of this paper is to present a general iterative scheme as below:

$$ \left\{ {\begin{array}{*{20}{c}} {F\left( {{u_n},y} \right) + \frac{1}{r_n}\left\langle {y - {u_n},{u_n} - {x_n}} \right\rangle \geqslant 0,} & {\forall y \in C,} \\ {{x_{n + 1}} = \left( {I - {\alpha_n}A} \right)S{u_n} + {\alpha_n}\gamma f\left( {x_n} \right),} & {} \\ \end{array} } \right. $$

and to prove that, if {α n } and {r n } satisfy appropriate conditions, then iteration sequences {x n } and {u n } converge strongly to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of solution of a variational inequality, too. Furthermore, by using the above result, we can also obtain an iterative algorithm for solution of an optimization problem \( \mathop {\min }\limits_{x \in C} h(x) \), where h(x) is a convex and lower semicontinuous functional defined on a closed convex subset C of a Hilbert space H. The results presented in this paper extend, generalize and improve the results of Combettes and Hirstoaga, Wittmann, S.Takahashi, Giuseppe Marino, Hong-Kun Xu, and some others.

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References

  1. S. D. Flam and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program, 1997, 78(1): 29–41.

    Article  MathSciNet  Google Scholar 

  2. P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 2005, 6(1): 117–136.

    MATH  MathSciNet  Google Scholar 

  3. A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl., 2000, 241(1): 46–55.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 2004, 298(1): 279–291.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed points problems in Hilbert spaces, J. Math. Anal. Appl., 2007, 331(1): 506–515.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. K. Xu, Iterative algorithms for nonlinear operators, J. London. Math. Soc., 2002, 66(1): 240–256.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. K. Xu, An iterative approach to quadratic optimzation, J. Optim. Theory Appl., 2003, 116(3): 659–678.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Deutsch and I. Yamada, Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal. and Optimiz., 1998, 19(1/2): 33–56.

    MATH  MathSciNet  Google Scholar 

  9. I. Yamada, The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings, in Inherently Parallel Algorithm for Feasibility and Optimization (ed. by D. Butnariu, Y. Censor, and S. Reich), Elsevier, 2001, 473–504.

  10. I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces, Numer. Funct. Anal. Optim., 1998, 19(1): 165–190.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 2006, 318(1): 43–52.

    Article  MATH  MathSciNet  Google Scholar 

  12. Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc., 1967, 73: 561–597.

    Article  MathSciNet  Google Scholar 

  13. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  15. K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China

    Fang Zhang & Yongfu Su

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  1. Fang Zhang
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  2. Yongfu Su
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Correspondence to Fang Zhang.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant No. 10771050.

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Zhang, F., Su, Y. A general iterative method of fixed points for equilibrium problems and optimization problems. J Syst Sci Complex 22, 503–517 (2009). https://doi.org/10.1007/s11424-009-9182-6

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  • DOI: https://doi.org/10.1007/s11424-009-9182-6

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