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A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. (English) Zbl 1155.47318

Suppose that \(K\) is a nonempty closed convex subset of a real uniformly convex and smooth Banach space \(E\) with \(P\) as a sunny nonexpansive retraction. Let \(T_1,T_2:K\to E\) be two weakly inward and asymptotically nonexpansive mappings with respect to \(P\) with sequences \(\{k_n\},\{l_n\}\subset[1,\infty)\), \(\lim_{n\to\infty}k_n=1\), \(\lim_{n\to\infty}l_n=1\), and assume that \[ F(T_1)\cap F(T_2)=\{x\in K: T_1x=T_2x=x\}\neq\emptyset. \] Let \(\{x_n\}\) be the sequence in \(K\) generated iteratively by the rule \[ x_1\in K,\quad x_{n+1}=\alpha_nx_n+\beta_n(PT_1)^nx_n+\gamma_n(PT_2)^nx_n\;\forall n\geq1, \] where \(\{\alpha_n\}, \{\beta_n\}\), and \(\{\gamma_n\}\) are three real sequences in \([\varepsilon,1-\varepsilon]\) for some \(\varepsilon>0\) which satisfy the condition \(\alpha_n+\beta_n+\gamma_n=1\). Then we have the following.
(1) If one of \(T_1\) and \(T_2\) is completely continuous or demicompact and \(\sum_{n=1}^\infty(k_n-1)<\infty\), \(\sum_{n=1}^\infty(l_n-1)<\infty\), then \(\{x_n\}\) converges strongly to some \(q\in F(T_1)\cap F(T_2)\).
(2) If \(E\) is a real uniformly convex Banach space satisfying Opial’s condition or whose norm is Fréchet differentiable, then the weak convergence of \(\{x_n\}\) to some \(q\in F(T_1)\cap F(T_2)\) is proved.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

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