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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 $\left\{{k}_{n}\right\}$ with ${k}_{n}\ge 1$ and ${lim}_{n\to \infty }{k}_{n}=1$ such that $\parallel {T}^{n}x-{T}^{n}y\parallel \le {k}_{n}\parallel x-y\parallel$ for all $x,y\in K$ and for all $n\in ℕ$. 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 $F\left(T\right)\ne \varnothing$ and sequence $\left\{{k}_{n}\right\}\subset \left[1,\infty \right)$ such that $lim{k}_{n}=1$ and ${\sum }_{n=1}^{\infty }\left({k}_{n}-1\right)<\infty$. Let $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ be bounded sequences in $K$ and let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{\text{'}}\right\}$, $\left\{{b}_{n}^{\text{'}}\right\}$ and $\left\{{c}_{n}^{\text{'}}\right\}$ be real sequence in $\left[0,1\right]$ satisfying the conditions:

(i) ${a}_{n}+{b}_{n}+{c}_{n}={a}_{n}^{\text{'}}+{b}_{n}^{\text{'}}+{c}_{n}^{\text{'}}=1$, $\forall n\ge 1$;

(ii) $a<{a}_{n}<{b}_{n}^{\text{'}}, $\forall n\ge 1$;

(iii) $lim{b}_{n}=0$;

(iv) ${\sum }_{n=1}^{\infty }{e}_{n}<\infty$, ${\sum }_{n=1}^{\infty }{c}_{n}^{\text{'}}<\infty$.

Then the sequence generated from an arbitrary ${x}_{1}\subset K$ by ${y}_{n}={a}_{n}{x}_{n}+{b}_{n}{T}^{n}{x}_{n}+{c}_{n}{u}_{n}$, $n\ge 1$, ${x}_{n+1}={a}_{n}^{\text{'}}{x}_{n}+{b}_{n}^{\text{'}}{T}^{n}{y}_{n}+{c}_{n}^{\text{'}}{v}_{n}$, $n\ge 1$ 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 $F\left(T\right)\ne \varnothing$ and sequence $\left\{{k}_{n}\right\}\subset \left[1,\infty \right)$ such that $lim{k}_{n}=1$ and ${\sum }_{n=1}^{\infty }\left({k}_{n}-1\right)<\infty$. Suppose ${T}^{n}$ is compact for some $m\in ℕ$. Let $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ be bounded sequence in $K$ and let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$, $\left\{{a}_{n}^{\text{'}}\right\}$, $\left\{{b}_{n}^{\text{'}}\right\}$ and $\left\{{c}_{n}^{\text{'}}\right\}$ be as in Theorem 1. Then the sequence $\left\{{x}_{n}\right\}$ 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 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 46B20 Geometry and structure of normed linear spaces