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

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.