Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

Abstract

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.

Keywords:

• Fixed points
• Hadamard manifolds
• Halpern-type algorithm
• inclusions problems
• Mann-type algorithm
• maximal monotone vector fields
• nonexpansive mappings
• Riemannian metric

MATHEMATICS SUBJECT CLASSIFICATION:

• 49J53
• 58E35
• 47J22
• 58C30
• 47J25
• 49J40
• 47J20
• 47H05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

In this research, second author was supported by a research grant of DSR-SERB No. EMR/2016/005124.