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Weak and strong convergence of an iterative method for nonexpansive mappings in Hilbert spaces. (English) Zbl 1199.47279
Let \(H\) be a real Hilbert space. Starting from an arbitrary initial point \(x_0\in H\), an iterative process can be defined by \[ x_{n+1}=a_nx_n+(1-a_n)T^{\lambda_{n+1}}_fy_n,\enskip y_n=b_nx_n+(1-b_n)T^{\beta_n}_gx_n,\quad n\geq 0, \] where \(T^{\lambda_{n+1}}_fx=Tx-\lambda_{n+1}\mu_ff(Tx)\), \(T^{\beta_n}_gx=Tx-\beta_n\mu_gg(Tx)\) (\(x\in H\)); \(T:H\to H\) is a nonexpansive mapping with \(F(T)\neq\emptyset\) and \(f:H\to H\) (resp., \(g:H\to H\)) is an \(\eta_f\) (resp., \(\eta_g\))-strongly monotone and \(k_f\) (resp., \(k_g\))-Lipschitzian mapping, \(\{a_n\}\subset(0,1)\), \(\{b_n\}\subset(0,1)\) and \(\{\lambda_n\}\subset [0,1)\), \(\{\beta_n\}\subset[0,1)\). Under some suitable conditions, some results related to the weak and the strong convergence of the sequence \(\{x_n\}\) to a fixed point of \(T\) are presented.

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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