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A general iterative method for an infinite family of nonexpansive mappings. (English) Zbl 1223.47105
Summary: Let $$H$$ be a real Hilbert space. Consider the iterative sequence
$x_{n+1}=\alpha_n\gamma f(x_n)+\beta_nx_n+((1-\beta_n)I-\alpha_nA)W_nx_n,$
where $$\gamma>0$$ is some constant, $$f:H\to H$$ is a given contractive mapping, $$A$$ is a strongly positive bounded linear operator on $$H$$ and $$W_n$$ is the $$W$$-mapping generated by an infinite countable family of nonexpansive mappings $$T_1,T_2,\dots,T_n,\dots$$ and $$\lambda_1,\lambda_2,\dots,\lambda_n,\dots$$ such that the common fixed points set $$F:=\bigcap^\infty_{n=1}\text{Fix}(T_n)\neq\emptyset$$. Under very mild conditions on the parameters, we prove that $$\{x_n\}$$ converges strongly to $$p\in F$$ where $$p$$ is the unique solution in $$F$$ of the following variational inequality: $$\langle(A-\gamma f)p,p-x^*\rangle\leq 0$$ for all $$x^*\in F$$, which is the optimality condition for the minimization problem
$\min_{x\in F}\tfrac12 \langle Ax,x \rangle-h(x).$

MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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