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Strong convergence theorems for semigroups of asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1483.47116

Summary: Let \(X\) be a real reflexive Banach space with a weakly continuous duality mapping \(J_\varphi\). Let \(C\) be a nonempty weakly closed star-shaped (with respect to \(u\)) subset of \(X\). Let \(\mathcal{F} = \{T(t) : t \in [0, +\infty]\}\) be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of \(C\), which is uniformly continuous at zero. We will show that the implicit iteration scheme \(y_n = \alpha_n u + (1 - \alpha_n)T(t_n)y_n\), for all \(n \in \mathbb N\), converges strongly to a common fixed point of the semigroup \(\mathcal F\) for some suitably chosen parameters \(\{\alpha_n\}\) and \(\{t_n\}\). Our results extend and improve corresponding ones of T. Suzuki [Proc. Am. Math. Soc. 131, No. 7, 2133–2136 (2003; Zbl 1031.47038)], H.-K. Xu [Bull. Aust. Math. Soc. 72, No. 3, 371–379 (2005; Zbl 1095.47016)], and H. Zegeye and N. Shahzad [Numer. Funct. Anal. Optim. 30, No. 7–8, 833–848 (2009; Zbl 1177.47084)].

MSC:

47J26 Fixed-point iterations
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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