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Strong convergence theorem of implicit iteration process for generalized asymptotically nonexpansive mappings in Hilbert space. (English) Zbl 1157.47047
Summary: Let \(C\) be a nonempty closed and convex subset of a Hilbert space \(H\), and let \(T\) and \(S:C\rightarrow C\) be two commutative generalized asymptotically nonexpansive mappings. We introduce an implicit iteration process of \(S\) and \(T\) defined by \(x_{n}=\alpha _{n}x_{0}+(1 - \alpha_{n})(2/((n+1)(n+2)))\sum _{k=0}^{n}\sum _{i+j=k}S^{i}T^{j}x_{n}\), and then prove that \(x_{n}\) converges strongly to a common fixed point of \(S\) and \(T\).
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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