A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces


The purpose of this paper is to propose and analyze a multi-step iterative sequence to solve a convex optimization problem and a fixed point problem in an Hadamard space. We aim to establish strong and \( \triangle \)-convergence results of the proposed iterative sequence by employing suitable conditions on the control parameters and the structural properties of the underlying space. As a consequence, we compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically quasi-nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results in the existing literature.

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The authors are very grateful to the editor and anonymous referees for their helpful comments.

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All authors have contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Correspondence to Muhammad Aqeel Ahmad Khan.

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The author M.A.A. Khan was supported by the Higher Education Commission of Pakistan through project No. NRPU 5332. The author P. Cholamjiak was supported by Thailand Research Fund and University of Phayao under Grant no. RSA 6180084 and Unit of Excellence, University of Phayao.

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Khan, M.A.A., Cholamjiak, P. A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces. J. Fixed Point Theory Appl. 22, 62 (2020). https://doi.org/10.1007/s11784-020-00796-3

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  • Convex optimization
  • lower semicontinuity
  • proximal point algorithm
  • total asymptotically quasi-nonexpansive mapping
  • common fixed point
  • asymptotic center

Mathematics Subject Classification

  • 47H09
  • 47H10
  • 65K10
  • 65K15