Some comparison theorems for weak nonnegative splittings of bounded operators

Abstract

AbstractThe comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors. In this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general Banach space and rather general cones, and in a Hilbert space, which extend some of the results obtained by Woźnicki (Japan J. Indust. Appl. Math. 11(1994) 289–342) and Marek and Szyld (Numer. Math. 44(1984) 23–35). Furthermore, we present new theorems also for bounded operator which extend some results by Csordas and Varga (Numer. Math. 44. (1984) 23–35) for weak nonnegative splittings of matrices