On deterministic convergence of iterations of relaxed projection operators

Abstract In many signal-processing and system-theoretic applications, such as image restoration [9, 10, 15–18] and adaptive control [1–3, 6–8], it is necessary to consider the convergence properties of iterations of relaxed projection operators drawn from either a finite or an infinite pool. Moreover, in an adaptive environment it is invariably true that the choice of iteration sequence is unknown or totally unrestricted. Unfortunately, relatively little is known about this subject, especially if the operators are nonlinear and the setting is infinite-dimensional. In addition to summarizing some of the more pertinent recent literature, we also present some new results which can serve as a starting point for further study. A fairly extensive discussion of ideas and theorems is followed by detailed proofs.