Strong convergence theorems of $k$-strict pseudo-contractions in Hilbert spaces

Qin, Xiaolong ; Kang, Shin Min ; Shang, Meijuan

Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 695-706.

Résumé

Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $T\: K\rightarrow H$ a $k$-strict pseudo-contraction for some $0\leq k1$ such that $F(T)=\{x\in K\: x=Tx\}\neq \emptyset $. Consider the following iterative algorithm given by $$ \forall x_1\in K,\quad x_{n+1}=\alpha _n\gamma f(x_n)+\beta _nx_n+((1-\beta _n)I-\alpha _n A)P_KSx_n,\quad n\geq 1, $$ where $S\: K\rightarrow H$ is defined by $Sx=kx+(1-k)Tx$, $P_K$ is the metric projection of $H$ onto $K$, $A$ is a strongly positive linear bounded self-adjoint operator, $f$ is a contraction. It is proved that the sequence $\{x_n\}$ generated by the above iterative algorithm converges strongly to a fixed point of $T$, which solves a variational inequality related to the linear operator $A$. Our results improve and extend the results announced by many others.

Détail

Classification : 47H09, 47H10, 47J25
Keywords: Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point