New shrinking iterative methods for infinite families of monotone operators in a Banach space, computational experiments and applications

Li Wei,Yanan Zhang,Rui Chen,Ravi P Agarwal

doi:10.1186/s13660-020-02330-y

Abstract

New shrinking iterative algorithms for approximating common zeros of two infinite families of maximal monotone operators in a real uniformly convex and uniformly smooth Banach space are designed. Two steps of multiple choices can be made in the new iterative algorithms, two groups of interactive containment sets C_{n} and Q_{n} are constructed and computational errors are considered, which are different from the previous ones. Strong convergence theorems are proved under mild assumptions and some new proof techniques can be found. Computational experiments for some special cases are conducted to show the effectiveness of the iterative algorithms and meanwhile some inequalities are proved to guarantee the strong convergence. Moreover, the applications of the abstract results on convex minimization problems and variational inequalities are exemplified.

Highlights

Throughout this paper, suppose E is a real Banach space with E∗ being its dual space
For a nonlinear mapping S : D(S) ⊂ E → 2E∗, we use S–10 to denote the set of zeros of S, that is, S–10 = {x ∈ D(S) : 0 ∈ Sx}
Two steps of multiple choices can be made in the new iterative algorithms and two groups of interactive containment sets Cn and Qn are constructed, which are different from the previous ones(e.g. [18])

Summary

Introduction

Throughout this paper, suppose E is a real Banach space with E∗ being its dual space. If E is a real smooth and strictly convex Banach space, there exists a unique sunny generalized non-expansive retraction of E onto C, which is denoted by RC. The iterative algorithm is presented in a real smooth and uniformly convex Banach space E as follows:

Results

Conclusion