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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References:

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