| This paper deals with a viscosity-like method for approximating a specific solution of the following fixed-point problem: find $\xt \in {\cal H}; \ \xt = (proj_{Fix(T)} \circ P) \xt$, where ${\cal H}$ is a Hilbert space, $P$ and $T$ are two nonexpansive mappings on a closed convex subset $D$ and $proj_{ Fix(T)}$ denotes the metric projection on the set of fixed-points of $T$. This amounts to saying that $\xt$ is the fixed-point of $T$ which satisfies a variational inequality depending on a given criterion $P$, namely: find $\xt \in {\cal H}; \ 0\in (I-P)\xt+N_{ Fix(T)}\xt$, where $N_{ Fix(T)}$ denotes the normal cone to the set of fixed-points of $T$. Strong convergence results for the viscosity-like method are proved. It should be noticed that the proposed method can be regarded, for instance, as a generalized version of Halpern's algorithm. |
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