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

Abstract

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 $D(B)\subset H$: $x_{n+\mathrm{1}}={\alpha }_{n}f(x_n)+{\gamma }_n{x}_n+{\delta }_n(I+r_{n}B{)}^{-\mathrm{1}}(I-r_{n}A)x_n+e_n$, for $n=\mathrm{1,2},\dots ,$ for given $x_{\mathrm{1}}$ in a real Hilbert space $H$, where $({\alpha }_n)$, $({\gamma }_n)$, and $({\delta }_n)$ are sequences in $(\mathrm{0,1})$ with ${\alpha }_n+{\gamma }_n+{\delta }_n=\mathrm{1}$ for all $n\ge \mathrm{1}$, $(e_n)$ denotes the error sequence, and $f:H\to H$ is a contraction. The algorithm is known to converge under the following assumptions on ${\delta }_n$ and $e_n$: (i) $({\delta }_n)$ is bounded below away from 0 and above away from 1 and (ii) $(e_n)$ is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i) $({\delta }_n)$ is bounded below away from 0 and above away from 3/2 and (ii) $(e_n)$ is square summable in norm; and we still obtain strong convergence results.