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Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. (English) Zbl 1163.47051

Summary: We introduce an implicit iteration scheme with a perturbed mapping for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space. Then, we establish some convergence theorems for this implicit iteration scheme which are connected with results by H. K. Xu and R. G. Ori [Numer. Funct. Anal. Optimization 22, No. 5–6, 767–773 (2001; Zbl 0999.47043)], L.–C. Zeng and J.–C. Yao [Nonlinear Anal., Theory Methods Appl. 64, No. 11 (A), 2507–2515 (2006; Zbl 1105.47061)] and S. Takahashi and W. Takahashi [J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)]. In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J40 Variational inequalities
91A25 Dynamic games
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