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Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications. (English) Zbl 1162.47051

Summary: We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically \(\kappa \)-strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to A.Tada and W.Takahashi [J. Optim.Theory Appl.133, No.3, 359–370 (2007; Zbl 1147.47052)] and K.Nakajo, K.Shimoji and W.Takahashi [Taiwanese J. Math.10, No.2, 339–360 (2006; Zbl 1109.47060)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
90C30 Nonlinear programming
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