# zbMATH — the first resource for mathematics

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.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text: