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Weak and strong convergence of a new scheme for two non-expansive mappings in Hilbert spaces. (English)
Appl. Anal. 87, No. 5, 567-574 (2008).
Summary: In a real Hilbert space $H$, from an arbitrary initial point $x_{0} \in H$, an iterative process is defined as follows: $x_{n+1} = a_{n}x_{n} + (1 - a_{n})T^{λ_{n+1}}_{f}y_{n}$, $y_{n} = b_{n}x_{n} + (1 - b_{n})S^{β_{n}}_{g}x_{n}$, $n \geq 0$, where $T^{λ_{n+1}}_{f}x = Tx - λ_{n+1}μ_{f}f(Tx)$, $S^{β_{n}}_{g}x = Sx - β_{n}μ_{g}g(sx)$, ($\forall x \in H$), $T, S : H \to H$ are two non-expansive mapping with $F(T) \cap F(S) \neq \emptyset$ and $f$ (resp. $g$) $: H \to H$ an $η_{f}$ (resp. $η_{g}$)-strongly monotone and $k_{f}$ (resp. $k_{g}$)-Lipschitzian mapping, $\{a_{n}\} \subset (0, 1), \{b_{n}\} \subset (0, 1)$ and $\{λ_{n}\} \subset [0, 1), \{β_{n}\} \subset [0, 1)$. Under some suitable conditions, several convergence results of the sequence $\{x_{n}\}$ are shown.