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Strong convergence theorem for asymptotically nonexpansive mappings. (English) Zbl 0861.47030
In the present paper the authors prove the following Theorem: Let $C$ be a nonempty closed convex subset of a real Hilbert space and $T:C\to C$ be an asymptotically nonexpansive mapping (i.e. for each $n\geq 1$ there exists a real number $k_n\geq 1$ such that $|T^nx- T^ny|\leq k_n|x-y|$, $x,y\in C$ and $\lim_{n\to\infty} k_n=1$). Suppose that the set of fixed points of $T$, $F(T)$ is nonempty. Let $$b_n= {1\over n}\sum^n_{j=1} (1+|1-k_j|+ e^{-j}), \qquad 0<a<1, \quad x_0\in C.$$ Then the mapping $T_n$ on $C$ given by $$T_nx= a_n x_0+ (1-a_n) A_nx\qquad \text{for all }x\in C$$ where $$a_n= {{b_n-1} \over {b_n-1+a}} \qquad \text{and} \qquad A_n= {1\over n}\sum^n_{j=1} T^j,$$ has a unique fixed point $u_n$ in $C$ and the sequence $\{u_n\}$ converges strongly to the element of $F(T)$ which is nearest to $x_0$.

##### MSC:
 47H09 Mappings defined by “shrinking” properties 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations)
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##### References:
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