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

Oganeditse Aaron Boikanyo

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 operatorAand maximal monotone operatorsBwithD(B)⊂H:xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, forn=1,2,…,for givenx1in a real Hilbert spaceH, where(αn),(γn), and(δn)are sequences in(0,1)withαn+γn+δn=1for alln≥1,(en)denotes the error sequence, andf:H→His a contraction. The algorithm is known to converge under the following assumptions onδnanden: (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.

Highlights

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, and are sequences in (0, 1) with αn + γn + δn = 1 for all n ≥ 1, 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) is bounded below away from 0 and above away from 1 and (ii) is summable in norm. We show that these conditions can further be relaxed to, respectively, the following: (i) is bounded below away from 0 and above away from 3/2 and (ii) is square summable in norm; and we still obtain strong convergence results
Let H be a real Hilbert space endowed with the inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖

Summary

Introduction

Let H be a real Hilbert space endowed with the inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖. Where (rn) is a sequence of positive numbers, A and B are maximal monotone operators with D(B) ⊂ D(A), and A is single valued. Given an operator T : H → H, we say that I − T is demiclosed at zero if, for any sequence (xn) in H, the following implication holds: xn ⇀ x } 󳨐⇒ x ∈ Fix (T) .

Results

Conclusion