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Weak and strong convergence theorems by Mann’s type iteration and the hybrid method in Hilbert spaces. (English) Zbl 1065.47079

Let \(C\) be a nonempty closed convex subset of a real Hilbert space and \(\{T_n\}\) be a family of mappings of \(C\) into itself such that the set of all common fixed points of \(\{T_n\}\) is nonempty. The authors consider the sequence \(\{x_n\}\) generated by \(x_1= x\in C\), \(y_n\approx T_n x_n\), \(x_{n+1}= P_C(\alpha_n x_n+ (1-\alpha_n)y_n)\) \((\forall n\in\mathbb{N})\), where \(\{\alpha_n\}\subset(-\infty, 1]\) and \(\| y_n- T_n x_n\|< \varepsilon_n\) \((\forall n\in\mathbb{N})\). Then, they give the conditions on \(\{T_n\}\), \(\{\alpha_n\}\) and \(\{\varepsilon_n\}\) under which \(\{x_n\}\) converges weakly to a common fixed point of \(\{T_n\}\). Further, they consider the sequence \(\{x_n\}\) generated by the hybrid method in mathematical programming and give the conditions of \(\{T_n\}\) under which \(\{x_n\}\) converges strongly to a common fixed point of \(\{T_n\}\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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