-class algorithms for pseudocontractions and -strict pseudocontractions in Hilbert spaces

In this paper we study iterative algorithms for finding a common element of the set of fixed points of κ-strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu, Iterative methods for strict pseudo-contractions in hilbert spaces, Nonlinear Analysis 67 (2007) 2258–2271] and [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and lipschitz-continuous monotone mappings, SIAM Journal on Optimization 16 (4) (2006) 1230–1241] but our aim here is to provide the links between these known algorithms and the general framework of -class algorithms studied in [H.H. Bauschke, P.L. Combettes, A weak-to-strong convergence principle for fejér-monotone methods in hilbert spaces, Mathematics of Operations Research 26 (2) (2001) 248–264].