A Strong Convergence Theorem for a Finite Family of Bregman Demimetric Mappings in a Banach Space under a New Shrinking Projection Method

Bijan Orouji,Ravi P Agarwal,Ebrahim Soori,Donal O’Regan,Humberto Rafeiro

doi:10.1155/2021/9551162

Abstract

In this paper, using a new shrinking projection method and new generalizedk-demimetric mappings, we consider the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregmank-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space. To the best of our knowledge, such a theorem for Bregmank-demimetric mapping is the first of its kind in a Banach space. This manuscript is online on arXiv by the link http://arxiv.org/abs/2107.13254.

Highlights

Let H be a Hilbert space and let C be a nonempty, closed, and convex subset of H
We get that kUx − xk2 + kx − pk2 + 2hUx − x, x − pi ≤ kx − pk2 + tkx − Uxk2: ð4Þ
We get that kUx − Uyk2 ≤ kx − yk2 + tkx − Ux − ðy − UyÞk2, ð1Þ

Summary

Introduction

Let H be a Hilbert space and let C be a nonempty, closed, and convex subset of H. Let E be a reflexive real Banach space and C be a nonempty, closed, and convex subset of E. Let x ∈ int dom f , and the subdifferential of f at x is the convex mapping set ∂f : E ⟶ 2E∗ defined by

Results

Conclusion