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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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Authors and Affiliations

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

    Fang Zhang & Yongfu Su

Authors
  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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