## 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.

### Citations:

Zbl 1031.47038; Zbl 1095.47016; Zbl 1177.47084
Full Text:

### References:

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