The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operator A and maximal monotone operators B with D(B)⊂H: xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, for n=1,2,., for given x1 in a real Hilbert space H, where (αn), (γn), and (δn) are sequences in (0,1) with αn+γn+δn=1 for all n≥1, (en) denotes the error sequence, and f:H→H is a contraction. The algorithm is known to converge under the following assumptions on δn and en: (i) (δn) is bounded below away from 0 and above away from 1 and (ii) (en) is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i) (δn) is bounded below away from 0 and above away from 3/2 and (ii) (en) is square summable in norm; and we still obtain strong convergence results.