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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)].

(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. |

### Keywords:

asymptotically nonexpansive mapping; asymptotically pseudo-contractive mapping; modified Isikawa iterative sequence with errors; modified Mann iterative sequence with error; fixed point; normalized duality mapping; asymptotically nonexpansive; asymptotically pseudo-contractive; Ishikawa and Mann iterative sequence with errors
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\textit{S. S. Chang}, Proc. Am. Math. Soc. 129, No. 3, 845--853 (2001; Zbl 0968.47017)

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