Viscosity approximation methods for generalized equilibrium problems and fixed point problems |
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| Authors: | Lu-Chuan Ceng Qamrul Hasan Ansari Jen-Chih Yao |
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| Institution: | (1) Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China;(2) Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, P.O. Box 1169, Dhahran, 31261, Saudi Arabia;(3) Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India;(4) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, ROC |
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| Abstract: | The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized
equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces.
By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP.
On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding
a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is
proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions
of GEP and the set of fixed points of a nonexpansive mapping. |
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| Keywords: | Viscosity approximation method Generalized equilibrium problem Fixed points Nonexpansive mappings Strong convergence |
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