Abstract
In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more.
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The authors would like to thank the editor and referees for useful comments and suggestions.
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The second author is supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India, through grant 09/013(0584)/2015-EMR-I.
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Sahu, D.R., Kumar, A. & Kang, S.M. Proximal point algorithms based on S-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications. Numer Algor 86, 1561–1590 (2021). https://doi.org/10.1007/s11075-020-00945-2
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DOI: https://doi.org/10.1007/s11075-020-00945-2
Keywords
- Convex minimization problem
- Fixed point problem
- Nearly asymptotically quasi-nonexpansive mapping
- Mean nonexpansive mapping
- S-iteration process
- Proximal point algorithm
- CAT(0) space
- △-convergence