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2016 The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators
Oganeditse Aaron Boikanyo
Abstr. Appl. Anal. 2016: 1-10 (2016). DOI: 10.1155/2016/2371857

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)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 n1, (en) denotes the error sequence, and f:HH 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.

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Oganeditse Aaron Boikanyo. "The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators." Abstr. Appl. Anal. 2016 1 - 10, 2016. https://doi.org/10.1155/2016/2371857

Information

Received: 8 September 2015; Accepted: 8 December 2015; Published: 2016
First available in Project Euclid: 13 April 2016

zbMATH: 06929354
MathSciNet: MR3484217
Digital Object Identifier: 10.1155/2016/2371857

Rights: Copyright © 2016 Hindawi

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