《1979 Ergodic convergence to a zero of the sum of monotone operators in Hilbert space》.pdf

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 72, 383-390 (1979) Ergodic Convergence to a Zero of the Sum of Monotone Operators in Hilbert Space GREGORYB. PASTY Department of Mathematics, University of Southern Cafornia, Los Angeles, California 90007 Submitted by Ky Fan 1. INTRODUCTION Let H be a real Hilbert space, and A a maximal monotone operator on H with domain D(A) and range R(A). Work by Rockafellar [9] and BrCzis and Lions [3] has inspired recent investigation into convergence of the resolvent iteration (RI) x,,EH; x ?z= (I + 4-l X,-l for n 3 1 when 0 E R(A), where {} is a sequence of positive reals satisfying C h, = + 00. While this hypothesis on (A,} is in general not sufficient to guarantee even the weak convergence of {xJ, it is noted in [3] that C h, = + co does ensure the weak convergence of the sequence of weighted averages z, := 5 A Ai . (1) i=l i=l Here we examine several different extensions of (RI) involving more than one maximal monotone operator and containing both forward and backward steps. In each case, the general result is the weak convergence of {z,} to