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Modified Proximal Point Methods Involving Quasi-pseudocontractive Mappings in Hadamard Spaces

Abstract

In this paper, we propose two new proximal point methods involving quasi-pseudocontractive mappings in Hadamard spaces. We prove that the first method converges strongly to a common solution of a finite family of minimization problems and fixed point problem for a finite family of quasi-pseudocontractive mappings in an Hadamard space. We then extend this method to a more general method involving multivalued monotone operators to approximate the solution of monotone inclusion problem, which is an important optimization problem. We establish that this method converges strongly to a common zero of a finite family of multivalued monotone operators which is also a common fixed point of a finite family of quasi-pseudocontractive mappings in an Hadamard space. Furthermore, we provide various nontrivial numerical implementations of our method in Hadamard spaces (which are non-Hilbert) and compare them with some other recent methods in the literature.

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Acknowledgements

The authors sincerely thank the reviewer for his careful reading, constructive comments and fruitful suggestions that improved the manuscript.

Funding

The first author acknowledges the doctoral bursary and financial support from the University of KwaZulu-Natal, Durban, South Africa. The second author acknowledges the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The research of the third author is wholly supported by the National Research Foundation (NRF) South Africa (S& F-DSI/NRF Free Standing Postdoctoral Fellowship; Grant Number: 120784). The fourth author is supported by the NRF of South Africa Incentive Funding for Rated Researchers (Grant Number 119903).

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Ogwo, G.N., Abass, H.A., Izuchukwu, C. et al. Modified Proximal Point Methods Involving Quasi-pseudocontractive Mappings in Hadamard Spaces. Acta Math Vietnam (2022). https://doi.org/10.1007/s40306-022-00480-3

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  • DOI: https://doi.org/10.1007/s40306-022-00480-3

Keywords

  • Minimization problem
  • Monotone inclusion problem
  • Convex functions
  • Monotone operators
  • Quasi-pseudocontractive mappings
  • Hadamard space

Mathematics Subject Classification (2010)

  • 47H09
  • 47H10
  • 49J20
  • 49J40