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

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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 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.

Original languageEnglish
Article number2371857
JournalAbstract and Applied Analysis
Volume2016
DOIs
Publication statusPublished - Jan 1 2016

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Viscosity Approximation
Monotone Operator
Splitting Method
Viscosity
Norm
Maximal Monotone Operator
Hilbert spaces
Zero
Strong Convergence
Convergence Analysis
Convergence Results
Contraction
Hilbert space
Denote
Converge
Operator

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "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)⊂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.",
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N2 - 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.

AB - 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.

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