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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 $$E$$, $$\langle\cdot, \cdot\rangle$$ the dual pair between $$E$$ and $$E^*$$, $$D(T)$$, $$F(T)$$ 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 $$J(x)= \{f\in E^*:\langle x,f\rangle= \|x\|,\|f\|,\|f\|=\|x\|\}$$, $$x\in E$$. A mapping $$T: D(T)\subset E\to E$$ is said to be
(1) asymptotically nonexpansive if there exists a sequence $$\{k_n\}$$ in $$(0,\infty)$$ with $$\lim_{n\to\infty} k_n= 1$$ such that $$\|T^n x- T^ny\|\leq k_n\|x-y\|$$ for all $$x,y\in D(T)$$ and $$n= 1,2,\dots$$ ,
(2) asymptotically pseudo-contractive if there exists a sequence $$\{k_n\}$$ in $$(0,\infty)$$ with $$\lim_{n\to\infty} k=1$$ and for any $$x,y\in D(T)$$ there exists $$j(x- y)\in J(x- y)$$ such that $$\langle T^nx- T^ny, j(x- y)\rangle\leq k_n\|x-y\|^2$$ for all $$n= 1,2,\dots$$ .
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 $$T: D(T)\subset E\to E$$ be a mapping, let $$D(T)$$ be a nonempty convex subset of $$E$$, let $$x_0\in D(T)$$ be a given point, and let $$\alpha_n$$, $$\beta_n$$, $$\gamma_n$$ and $$\delta_n$$ be four sequences in $$[0,1]$$. Then the sequence $$\{x_n\}$$ defined by $$x_{n+1}= (1-\alpha_n- \gamma_n) x_n+ \alpha_nT^n y_n+ \gamma_n u_n$$, $$y_n= (1-\beta_n- \delta_n) x_n+ \beta_n T^nx_n+ \delta_n v_n$$ for all $$n\geq 0$$ is called the modifies Ishikawa iterative sequence with errors of $$T$$, where $$u_n$$ and $$v_n$$ are two bounded sequences in $$D(T)$$.
If $$\beta_n=0$$ and $$\delta_n= 0$$, $$n= 0,1,2,\dots$$, then $$y_n= x_n$$. The sequence $$x_{n+1}= (1- \alpha_n- \gamma_n) x_n+ \alpha_n T^n x_n+ \gamma_n u_n$$, $$n\geq 0$$, 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 $$\{k_n\}\in (0,\infty)$$, $$\lim_{n\to\infty}k_n= 1$$, and let $$F(T)\neq\emptyset$$. Let $$\alpha_n$$, $$\beta_n$$, $$\gamma_n$$, and $$\delta_n$$ be four sequences in $$[0,1]$$ satisfying the following conditions:
(i) $$\alpha_n+\gamma_n\leq 1$$, $$\beta_n+ \delta_n\leq 1$$;
(ii) $$\alpha_n\to 0$$, $$\beta_n\to 0$$, $$\delta_n\to 0$$ $$(n\to\infty)$$;
(iii) $$\sum^\infty_{n=0} \alpha_n=\infty$$, $$\sum^\infty_{n=0} \gamma_n<\infty$$.
Let $$x_0\in D$$ be any given point and let $$\{x_n\}$$, $$\{y_n\}$$ be the modified Ishikawa iterative sequence with errors. Then:
(1) If $$\{x_n\}$$ converges strongly to a fixed point $$q$$ of $$T$$ in $$D$$, there exists a nondecreasing function $$\emptyset:[0,\infty)\to [0,\infty)$$, $$\emptyset(0)= 0$$ such that $$\langle T^n y_n- q, J(y_n- q)\rangle\leq k_n\|y_n- q\|^2- \emptyset(\|y_n- q\|)$$, for all $$n\geq 0$$.
(2) Conversely, if there exists a strictly increasing function $$\emptyset: [0,\infty)\to [0,\infty)$$, $$\emptyset(0)= 0$$ satisfying preceding inequality, then $$x_n\to q\in F(T)$$.
A similar result for asymptotically nonexpansive mappings is proved. If $$\{x_n\}$$ 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 and generalizations 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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