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Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. (English) Zbl 1102.47055

Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\) and let \(T\) be a self-map of \(K\) such that \(F(T)\), the set of fixed points of \(T\), is nonempty. Then \(T\) is asymptotically quasi-nonexpansive with sequence \(\{v_n\}\subset [0, \infty]\) if \(\lim_{n\rightarrow \infty}= 0\) and \[ \| T^n x - x_*\| \leq\;(1 + v_n) \| x - x_*\| \] for all \(x\in K\), \(x_* \in F(T)\) and \(n \geq 1\).
The main purpose of this paper is to investigate conditions under which an Ishikawa type iteration scheme for two asymptotically quasi-nonexpansive self-maps \(S\) and \(T\) of \(K\) converges to a common fixed point of \(S\) and \(T\). The authors consider the case of the Ishikawa type iterations with erors as well. Some results of M. K. Ghosh and L. Debnath [J. Math. Anal. Appl. 207, No. 1, 96–103 (1997; Zbl 0881.47036)], Q.–H. Liu [J. Math. Anal. Appl. 259, No. 1, 1–7 (2001; Zbl 1033.47047)] and others are discussed as special cases.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

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