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Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings. (English) Zbl 0968.47017

Let $E$ be a real Banach space, ${E}^{*}$ the topological dual space of $E$, $〈·,·〉$ the dual pair between $E$ and ${E}^{*}$, $D\left(T\right)$, $F\left(T\right)$ the domain of $T$ and the set of all fixed points of $T$, respectively, and $J:E\to {2}^{{E}^{*}}$ the normalized duality mapping defined by $J\left(x\right)=\left\{f\in {E}^{*}:〈x,f〉=\parallel x\parallel ,\parallel f\parallel ,\parallel f\parallel =\parallel x\parallel \right\}$, $x\in E$. A mapping $T:D\left(T\right)\subset E\to E$ is said to be

(1) asymptotically nonexpansive if there exists a sequence $\left\{{k}_{n}\right\}$ in $\left(0,\infty \right)$ with ${lim}_{n\to \infty }{k}_{n}=1$ such that $\parallel {T}^{n}x-{T}^{n}y\parallel \le {k}_{n}\parallel x-y\parallel$ for all $x,y\in D\left(T\right)$ and $n=1,2,\cdots$ ,

(2) asymptotically pseudo-contractive if there exists a sequence $\left\{{k}_{n}\right\}$ in $\left(0,\infty \right)$ with ${lim}_{n\to \infty }k=1$ and for any $x,y\in D\left(T\right)$ there exists $j\left(x-y\right)\in J\left(x-y\right)$ such that $〈{T}^{n}x-{T}^{n}y,j\left(x-y\right)〉\le {k}_{n}{\parallel x-y\parallel }^{2}$ for all $n=1,2,\cdots$ .

In this paper some convergence theorems of modified Ishikawa and Mann iterative sequence with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach spaces are obtained.

Let $T:D\left(T\right)\subset E\to E$ be a mapping, let $D\left(T\right)$ be a nonempty convex subset of $E$, let ${x}_{0}\in D\left(T\right)$ be a given point, and let ${\alpha }_{n}$, ${\beta }_{n}$, ${\gamma }_{n}$ and ${\delta }_{n}$ be four sequences in $\left[0,1\right]$. Then the sequence $\left\{{x}_{n}\right\}$ defined by ${x}_{n+1}=\left(1-{\alpha }_{n}-{\gamma }_{n}\right){x}_{n}+{\alpha }_{n}{T}^{n}{y}_{n}+{\gamma }_{n}{u}_{n}$, ${y}_{n}=\left(1-{\beta }_{n}-{\delta }_{n}\right){x}_{n}+{\beta }_{n}{T}^{n}{x}_{n}+{\delta }_{n}{v}_{n}$ for all $n\ge 0$ is called the modifies Ishikawa iterative sequence with errors of $T$, where ${u}_{n}$ and ${v}_{n}$ are two bounded sequences in $D\left(T\right)$.

If ${\beta }_{n}=0$ and ${\delta }_{n}=0$, $n=0,1,2,\cdots$, then ${y}_{n}={x}_{n}$. The sequence ${x}_{n+1}=\left(1-{\alpha }_{n}-{\gamma }_{n}\right){x}_{n}+{\alpha }_{n}{T}^{n}{x}_{n}+{\gamma }_{n}{u}_{n}$, $n\ge 0$, is called the modified Mann iterative sequences with errors of $T$. Main result is the following:

Theorem 1. Let $E$ be a real uniformly smooth Banach space, let $D$ be a non-empty bounded closed convex subset of $E$, let $T:D\to D$ be an asymptotically pseudo-contractive mapping with a sequence $\left\{{k}_{n}\right\}\in \left(0,\infty \right)$, ${lim}_{n\to \infty }{k}_{n}=1$, and let $F\left(T\right)\ne \varnothing$. Let ${\alpha }_{n}$, ${\beta }_{n}$, ${\gamma }_{n}$, and ${\delta }_{n}$ be four sequences in $\left[0,1\right]$ satisfying the following conditions:

(i) ${\alpha }_{n}+{\gamma }_{n}\le 1$, ${\beta }_{n}+{\delta }_{n}\le 1$;

(ii) ${\alpha }_{n}\to 0$, ${\beta }_{n}\to 0$, ${\delta }_{n}\to 0$ $\left(n\to \infty \right)$;

(iii) ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$, ${\sum }_{n=0}^{\infty }{\gamma }_{n}<\infty$.

Let ${x}_{0}\in D$ be any given point and let $\left\{{x}_{n}\right\}$, $\left\{{y}_{n}\right\}$ be the modified Ishikawa iterative sequence with errors. Then:

(1) If $\left\{{x}_{n}\right\}$ converges strongly to a fixed point $q$ of $T$ in $D$, there exists a nondecreasing function $\varnothing :\left[0,\infty \right)\to \left[0,\infty \right)$, $\varnothing \left(0\right)=0$ such that $〈{T}^{n}{y}_{n}-q,J\left({y}_{n}-q\right)〉\le {k}_{n}\parallel {y}_{n}{-q\parallel }^{2}-\varnothing \left(\parallel {y}_{n}-q\parallel \right)$, for all $n\ge 0$.

(2) Conversely, if there exists a strictly increasing function $\varnothing :\left[0,\infty \right)\to \left[0,\infty \right)$, $\varnothing \left(0\right)=0$ satisfying preceding inequality, then ${x}_{n}\to q\in F\left(T\right)$.

A similar result for asymptotically nonexpansive mappings is proved. If $\left\{{x}_{n}\right\}$ is the modified Mann iterative sequence with errors, a similar result for asymptotically pseudo-contractive mappings is obtained.

This paper generalizes the results by [K. Goebel and W. A. Kirk, Proc. Am. Math. Soc. 35, No. 1, 171-174 (1972; Zbl 0256.47045), W. A. Kirk, Am. Math. Mont. 72, 1004-1006 (1965; Zbl 0141.32402), Q. H. Liu, Nonlinear Anal. Theory Methods Appl. 26, No. 11, 1835-1842 (1996; Zbl 0861.47047), J. Schu, J. Math. Anal. Appl. 15, No. 2, 407-413 (1991; Zbl 0734.47036)].

Reviewer: V.Popa (Bacau)

##### MSC:
 47H05 Monotone operators (with respect to duality) and generalizations 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties