Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

Kengo Kasahara,Yasunori Kimura

doi:10.3390/axioms10010015

Abstract

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.

Highlights

We prove convergence of Mann and Halpern types of iterative sequences for finitely many convex functions by using the properties of a sequence of the resolvents in CAT(1) space
We proposed a new type of iterative scheme for the problem of finding a common minimizer of finitely many convex functions defined on a complete CAT(1) space
We considered the resolvent operators for proper lower semicontinuous convex functions defined on a complete CAT(1) space and their convex combination

Summary

Introduction

We consider finding a common fixed point of a finite number of resolvents operators for proper lower semicontinuous convex functions on a geodesic space. To find this point, we often use iterative schemes. We know a large number of results by using Mann’s and Halpern’s iterative schemes in a CAT(1) space. Piatek [8] considered Halpern’s iterative scheme by using a nonexpansive mapping in CAT(1) space. Kimura and Kohsaka [11] proved convergence of Mann and Halpern types of iterative schemes with a sequence of resolvent operators for a single proper lower semicontinuous convex function.

Preliminaries

Lemmas for a Finite Number of Resolvent Operators

Iterative Schemes for a Finite Resolvents Operators

Applications to the Image Recovery Problem

Conclusions