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Keywords:
Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point
Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point
Summary:
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 $0\leq k<1$ 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.
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