  In this paper we present a systematic study of regular sequences of
quasi-nonexpansive operators in Hilbert space. We are interested, in
particular, in weakly, boundedly and linearly regular sequences of operators.
We show that the type of the regularity is preserved under relaxations, convex
combinations and products of operators. Moreover, in this connection, we show
that weak, bounded and linear regularity lead to weak, strong and linear
convergence, respectively, of various iterative methods. This applies, in
particular, to block iterative and string averaging projection methods, which,
in principle, are based on the above-mentioned algebraic operations applied to
projections. Finally, we show an application of regular sequences of operators
to variational inequality problems.
