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Convergence theorems for uniformly \(L\)-Lipschitzian asymptotically nonexpansive mappings. (English) Zbl 1222.47111

Summary: Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\), \(T:K\to K\) a uniformly \(L\)-Lipschitzian asymptotically pseudocontractive mapping with sequence \(\{k_n\}_{n\geq 0}\subset [1,\infty)\), \(\lim_{n\to\infty} k_n=1\) such that \(p\in F(T)=\{x\in K:Tx=x\}\). Let \(\{a_n\}_{n\geq 0}\), \(\{b_n\}_{n\geq 0}\), \(\{c_n\}_{n\geq 0}\) be real sequences in \([0,1]\) satisfying the following conditions: (i) \(a_n+b_n+c_n=1\); (ii) \(\sum_{n\geq 0} b_n=\infty\); (iii) \(c_n=o(b_n)\); (iv) \(\lim_{n\to\infty} b_n=0\).
For arbitrary \(x_0\in K\), let \(\{x_n\}_{n\geq 0}\) be iteratively defined by \[ x_{n+1}=a_nx_n+b_nT^nx_n+c_nu_n,\qquad n\geq 0, \]
where \(\{u_n\}_{n\geq 0}\) is a bounded sequence of error terms in \(K\). Suppose that there exists a strictly increasing function \(\psi:[0,\infty)\to [0,\infty)\), \(\psi(0)=0\), such that
\[ \langle T^nx-p,j(x-p)\rangle\leq k_n\| x-p\| ^2-\psi(\| x-p\|),\qquad \forall x\in K. \]
Then \(\{x_n\}_{n\geq 0}\) converges strongly to \(p\in F(T)\).
The results proved in this paper significantly improve the results of E. U. Ofoedu [J. Math. Anal. Appl. 321, No. 2, 722–728 (2006; Zbl 1109.47061)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1109.47061
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