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Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps. (English) Zbl 1126.65054

Let \(E\) be a real Banach space and \(K\) be a nonempty closed convex part of it. The map \(T:K\to K\) is called generalized asymptotically quasi-nonexpansive if \(F(T)\neq \emptyset\) and
\[ \| T^nx-x^*\| \leq (1+u_n)\| x-x^*\| +c_n,\quad n\geq 1,\;x\in K,\;x^*\in F(T), \] where \((u_n)\) and \((c_n)\) are sequences converging to zero. Let \((T_i; i=1,\dots,N)\) be a family of such maps. Sufficient conditions are given so that the implicit iterative scheme \[ x_n=\alpha_nx_{n-1}+(1-\alpha_n)T_i^kx_n, n\geq 1;\;x_0\in K \]
where \(n=(k-1)N+i\), \(T_n=T_{n\pmod N}=T_i\), \(i\in \{1,\dots,N\}\), should strongly converge to a common fixed point of the family \((T_i; i=1,\dots,N)\).
This, in particular, answers affirmatively a question raised by H.-K. Xu and R. G. Ori [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043)].

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 0999.47043
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References:

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