## Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings.(English)Zbl 0971.47038

Let $$X$$ be a uniformly convex Banach space and $$K$$ a nonempty subset of $$X$$. A mapping $$T: K\to K$$ is said to be asymptotically nonexpansive mapping if there exists a sequence $$\{k_n\}$$ with $$k_n\geq 1$$ and $$\lim_{n\to\infty} k_n=1$$ such that $$\|T^nx- T^ny\|\leq k_n\|x-y\|$$ for all $$x,y\in K$$ and for all $$n\in\mathbb{N}$$. In this paper, if $$K$$ is a nonempty closed convex subset of $$X$$ and $$T: K\to K$$ is a nonexpansive asymptotically mapping with a nonempty fixed point set, weak and strong convergence theorems for the iterative approximation of fixed points of $$T$$ are proved.
Furthermore, the results by this paper show that the boundedness requirement imposed on the subset $$K$$ in recent results by Z. Huang [Comput. Math. Appl. 37, No. 3, 1-7 (1999; Zbl 0942.47046)]; B. E. Rhoades [J. Mat. Anal. Appl. 183, No. 1, 118-120 (1994; Zbl 0807.47045)]; J. Schu [J. Math. Anal. Appl. 158, No. 2, 407-413 (1991; Zbl 0734.47036); Bull. Aust. Math. Soc. 43, No. 1, 153-159 (1991; Zbl 0709.47051)], can be dropped. The main results are the following:
Theorem 1: Let $$E$$ be a uniformly convex Banach space sastisfying Opial’s condition and let $$K$$ be a nonempty closed convex subset of $$E$$. Let $$T: K\to K$$ be an asymptotically nonexpansive mapping with $$F(T)\neq \emptyset$$ and sequence $$\{k_n\}\subset [1,\infty)$$ such that $$\lim k_n= 1$$ and $$\sum^\infty_{n=1} (k_n-1)< \infty$$. Let $$\{u_n\}$$ and $$\{v_n\}$$ be bounded sequences in $$K$$ and let $$\{a_n\}$$, $$\{b_n\}$$, $$\{c_n\}$$, $$\{a_n'\}$$, $$\{b_n'\}$$ and $$\{c_n'\}$$ be real sequence in $$[0,1]$$ satisfying the conditions:
(i) $$a_n+ b_n+ c_n= a_n'+ b_n'+ c_n'= 1$$, $$\forall n\geq 1$$;
(ii) $$a< a_n< b_n'< b< 1$$, $$\forall n\geq 1$$;
(iii) $$\lim b_n= 0$$;
(iv) $$\sum^\infty_{n=1} e_{n}<\infty$$, $$\sum^\infty_{n=1} c_n'< \infty$$.
Then the sequence generated from an arbitrary $$x_1\subset K$$ by $$y_n= a_n x_n+ b_n T^n x_n+ c_n u_n$$, $$n\geq 1$$, $$x_{n+1}= a_n' x_n+ b_n'T^n y_n+ c_n'v_n$$, $$n\geq 1$$ converges weakly to some fixed point of $$T$$.
Theorem 2. Let $$E$$ be a uniformly convex Banach space and $$K$$ a nonempty closed subset of $$E$$. Let $$T: K\to K$$ be an asymptotically nonexpansive mapping with $$F(T)\neq \emptyset$$ and sequence $$\{k_n\}\subset [1,\infty)$$ such that $$\lim k_n=1$$ and $$\sum^\infty_{n=1} (k_n-1)<\infty$$. Suppose $$T^n$$ is compact for some $$m\in\mathbb{N}$$. Let $$\{u_n\}$$ and $$\{v_n\}$$ be bounded sequence in $$K$$ and let $$\{a_n\}$$, $$\{b_n\}$$, $$\{c_n\}$$, $$\{a_n'\}$$, $$\{b_n'\}$$ and $$\{c_n'\}$$ be as in Theorem 1. Then the sequence $$\{x_n\}$$ generate from an arbitrary $$x_1\in K$$ as in Theorem 1 converges strongly to some fixed point of $$T$$.
Reviewer: V.Popa (Bacau)

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators 46B20 Geometry and structure of normed linear spaces

### Citations:

Zbl 0942.47046; Zbl 0807.47045; Zbl 0734.47036; Zbl 0709.47051
Full Text:

### References:

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