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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Ceng, LC., Ansari, Q.H. & Yao, JC. Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J Glob Optim 43, 487–502 (2009). https://doi.org/10.1007/s10898-008-9342-6
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DOI: https://doi.org/10.1007/s10898-008-9342-6