The viscosity approximation forward-backward splitting method for zeros of the sum of monotone operators. (English) Zbl 1470.65109

Summary: 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 \(D(B) \subset H\): \(x_{n + 1} = \alpha_n f(x_n) + \gamma_n x_n + \delta_n(I + r_n B)^{- 1}(I - r_n A) x_n + e_n\), for \(n = 1,2, \ldots\), for given \(x_1\) in a real Hilbert space \(H\), where \((\alpha_n)\), \((\gamma_n)\), and \((\delta_n)\) are sequences in \((0,1)\) with \(\alpha_n + \gamma_n + \delta_n = 1\) for all \(n \geq 1\), \((e_n)\) denotes the error sequence, and \(f : H \rightarrow H\) 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.


65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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