Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

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 abovementioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems.