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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (10): 1273-1282.doi: https://doi.org/10.1007/s10483-010-1360-x

• Articles • 上一篇    下一篇

Viscosity approximation with weak contractions for fixed point problem, equilibrium problem, and variational inequality problem

张石生1 李向荣2 陈志坚2 柳京爱3   

  1. 1. Department of Mathematics, Yibin University, Yibin 644007, Sichuan Province, P. R. China;
    2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, P. R. China;
    3. Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, P. R. China
  • 收稿日期:2010-05-03 修回日期:2010-09-08 出版日期:2010-10-01 发布日期:2010-10-01

Viscosity approximation with weak contractions for fixed point problem, equilibrium problem, and variational inequality problem

ZHANG Shi-Sheng1, LEE Heung-wing Joseph2, CHAN Chi-kin2, LIU Jing-Ai3   

  1. 1. Department of Mathematics, Yibin University, Yibin 644007, Sichuan Province, P. R. China;
    2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, P. R. China;
    3. Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, P. R. China
  • Received:2010-05-03 Revised:2010-09-08 Online:2010-10-01 Published:2010-10-01

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摘要： This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of Colao et al. (Colao, V., Acedo, G. L., and Marino, G. An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 71, 2708–2715 (2009)), Plubtieng and Punpaeng (Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007)), Colao et al. (Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium problem and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)), Yao et al. (Yao, Y., Liou, Y. C., and Yao, J. C. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Application 2007, Article ID 64363 (2007) DOI 10.1155/2007/64363), and others.

Abstract: This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of Colao et al. (Colao, V., Acedo, G. L., and Marino, G. An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 71, 2708–2715 (2009)), Plubtieng and Punpaeng (Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007)), Colao et al. (Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium problem and fixed point problems. J. Math. Anal. Appl. 344, 340–352 (2008)), Yao et al. (Yao, Y., Liou, Y. C., and Yao, J. C. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Application 2007, Article ID 64363 (2007) DOI 10.1155/2007/64363), and others.

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