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.