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Convergence to common fixed point for generalized asymptotically nonexpansive semigroup in Banach spaces. (English) Zbl 1177.47076
Let \(K\) be a nonempty closed convex subset of a reflexive and strictly convex Banach space \(E\) with a uniformly Gâteaux differentiable norm, \({\mathcal F}=\{T(h) :h\geq 0\}\) be a generalized asymptotically nonexpansive self-mapping semigroup of \(K\), and \(f:K\to K\) be a fixed contractive mapping with contractive coefficient \(\beta\in(0,1)\). The authors prove that the following implicit and modified implicit viscosity iterative schemes \(\{x_n\}\) defined by \(x_n=\alpha_nf(x_n)+(1-\alpha_n)T(t_n)x_n\) and \(x_n=\alpha_ny_n+(1-\alpha_n)T(t_n)x_n\), \(y_n=\beta_n f(x_{n-1})+(1-\beta_n)x_{n-1}\), strongly converge to \(p\in F\) as \(n\to\infty\), and \(p\) is the unique solution to the following variational inequality: \(\langle f(p)-p,j(y-p)\rangle\leq 0\) for all \(y\in F\).

MSC:
47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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