Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1102.47056

Let \(E\) be a real Banach space with uniformly Gâteaux differentiable norm, \(K\) a nonempty bounded closed convex subset of \(E\), and \(T: K \to K\) an asymptotically nonexpansive mapping, i.e., \(\| T^n x- T^n y \| \leq k_n \| x-y \|\) for all \(x, y \in K\), where \(k_n \in [1, \infty)\), \(\lim_{n \to \infty} k_n=1\). Let \(f: K \to K\) be a contraction. The authors study approximating properties of the sequence \(\{ x_n\}\) defined by \(x_n=(1-k_n^{-1} t_n) f(x_n)+k_n^{-1} t_n T^n x_n\) (\(t_n>0\), \(\lim_{n \to \infty} t_n=1\)) with respect to the set of fixed points of the mapping \(T\).


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H06 Nonlinear accretive operators, dissipative operators, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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