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Convergence theorem for \(I\)-asymptotically quasi-nonexpansive mapping in Hilbert space. (English) Zbl 1153.47309

Summary: Let \(\mathcal H\) be a Hilbert space with inner product (\(\cdot,\cdot\)) and \(\|\cdot\|\) norm, and let \(K\) be weakly compact a subset of \(\mathcal H\). Let \(T:K \to K\) be nonlinear mapping and \(I:K \to K\) be a nonlinear bounded mapping. In this paper, we define the \(I\)-asymptotically quasi-nonexpansive mapping in Hilbert space. If \(T\) is an \(I\)-asymptotically quasi-nonexpansive mapping, then we prove that \(\frac{1}{n}\sum^{n-1}_{i=0} T^i u\), for \(u \in K\) as \(n\rightarrow \infty \), is weakly almost convergent to its asymptotic center.

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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