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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 \(\{k_n\}\) 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 \(T\).

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 |

### Keywords:

uniformly convex Banach space; asymptotically nonexpansive mapping; iterative approximation of fixed points; Opial’s condition
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\textit{M. O. Osilike} and \textit{S. C. Aniagbosor}, Math. Comput. Modelling 32, No. 10, 1181--1191 (2000; Zbl 0971.47038)

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### References:

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