On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

Grace N Ogwo,Oluwatosin T Mewomo,Kazeem O Aremu,Chinedu Izuchukwu

doi:10.1515/dema-2020-0006

Grace N Ogwo, Oluwatosin T Mewomo + Show 2 more

Open Access

https://doi.org/10.1515/dema-2020-0006

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Journal: Demonstratio Mathematica	Publication Date: Jul 3, 2020
Citations: 13	License type: CC BY 4.0

Affiliation: University of KwaZulu-Natal

Abstract

AbstractOur main interest in this article is to introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.

Highlights

Let X be a metric space and C be a nonempty subset of X
A Halperntype proximal point algorithm (PPA) comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space
We apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems (VIPs) and convex feasibility problems in Hadamard spaces

Summary

Introduction

Let X be a metric space and C be a nonempty subset of X. They gave an example of a demimetric mapping and established some basic results for this class of mappings They proved a strong convergence theorem for approximating a fixed point of demimetric mappings and a solution to a minimization problem in Hadamard spaces. Motivated by the aforementioned results and the current interest in this research direction, we introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. A Halperntype PPA comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. We apply the obtained results to solve a finite family of convex minimization problems, VIPs and convex feasibility problems in Hadamard spaces

Preliminaries

Main results

Application to some optimization problems

Convex feasibility problem

Convex minimization problem

Conclusion