A simultaneous iterative method for split equality problems of two finitefamilies of strictly pseudononspreading mappings without prior knowledge ofoperator norms

Haitao Che,Meixia Li

doi:10.1186/1687-1812-2015-1

Abstract

AbstractIn this article, we first introduce the concept of T-mapping of a finitefamily of strictly pseudononspreading mapping "Equation missing", and we show that the fixed point set of theT-mapping is the set of common fixed points of"Equation missing" and T is a quasi-nonexpansive mapping.Based on the concept of a T-mapping, we propose a simultaneousiterative algorithm to solve the split equality problem with a way of selectingthe stepsizes which does not need any prior information about the operatornorms. The sequences generated by the algorithm weakly converge to a solution ofthe split equality problem of two finite families of strictly pseudononspreadingmappings. Furthermore, we apply our iterative algorithms to some convex andnonlinear problems.MSC: 47H09, 47H05, 47H06, 47J25.

Highlights

Due to their extraordinary utility and broad applicability in many areas of applied mathematics, algorithms for solving convex feasibility problems continue to receive great attention; see for instance [ – ]
In intensity-modulated radiation therapy (IMRT), this amounts to envisage a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity, for further details, the interested reader is referred to [, ]
We introduce a new choice of the stepsize sequence {γk} for the simultaneous iterative algorithm to solve ( . ) governed by quasi-nonexpansive mapping as follows: γk ∈, min

Summary

Introduction

Due to their extraordinary utility and broad applicability in many areas of applied mathematics (most notably, fully discretized models of problems in image reconstruction from projections, in image processing, and in intensity-modulated radiation therapy), algorithms for solving convex feasibility problems continue to receive great attention; see for instance [ – ]. B, which is in general not easy in practice To overcome this difficulty, Lopez et al [ ] and Zhao et al [ ] presented useful method for choosing the stepsizes which do not need prior knowledge of the operator norms for solving the split feasibility problems and multiple-set split feasibility problems, respectively. We introduce a new choice of the stepsize sequence {γk} for the simultaneous iterative algorithm to solve We propose the following simultaneous iterative algorithm where the stepsizes do not depend on the operator norms A and B and prove the weak convergence of the algorithm to solve

Assume that

Notice that

It yields