## 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 $$T$$.

### MSC:

 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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### References:

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