Skip to main content Accessibility help
×
Home

$\unicode[STIX]{x1D6E5}$ -CONVERGENCES OF WEIGHTED AVERAGED PROJECTIONS IN $\text{CAT}(\unicode[STIX]{x1D705})$ SPACES

  • BYOUNG JIN CHOI (a1)

Abstract

We first introduce the weighted averaged projection sequence in $\text{CAT}(\unicode[STIX]{x1D705})$ spaces and then we establish some inequalities for the weighted averaged projection sequence. Using the inequalities, we prove the asymptotic regularity and the $\unicode[STIX]{x1D6E5}$ -convergence of the weighted averaged projection sequence. Furthermore, we prove the strong convergence of the sequence under certain regularity or compactness conditions on $\text{CAT}(\unicode[STIX]{x1D705})$ spaces.

Copyright

Footnotes

Hide All

The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2017R1C1B1005334) and the 2020 scientific promotion program funded by Jeju National University.

Footnotes

References

Hide All
[1]Ariza-Ruiz, D., López-Acedo, G. and Nicolae, A., ‘The asymptotic behavior of the composition of firmly nonexpansive mappings’, J. Optim. Theory Appl. 167 (2015), 409429.
[2]Auslender, A., ‘Méthodes numériques pour la résolution des problèmes d’optimisation avec constraintes’, Thèse, Faculté des Sciences, Université de Grenoble, 1969.
[3]Bačák, M., Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 22 (De Gruyter, Berlin, 2014).
[4]Bačák, M., Searston, I. and Sims, B., ‘Alternating projections in CAT(0) spaces’, J. Math. Anal. Appl. 385 (2012), 599607.
[5]Bauschke, H. H. and Borwein, J. M., ‘On projection algorithms for solving convex feasibility problems’, SIAM Rev. 38 (1996), 367426.
[6]Bauschke, H. H., Matoušková, E. and Reich, S., ‘Projection and proximal point methods: convergence results and counterexamples’, Nonlinear Anal. 56 (2004), 715738.
[7]Bhatia, R., Positive Definite Matrices, Princeton Series in Applied Mathematics (Princeton University Press, Princeton, NJ, 2007).
[8]Brègman, L. M., ‘Finding the common point of convex sets by the method of successive projection’, Dokl. Akad. Nauk SSSR 162 (1965), 487490; (in Russian); English transl. in Sov. Math. Dokl. 6 (1965), 688–692.
[9]Bridson, M. R. and Haefliger, A., Metric Spaces of Non-positive Curvature, Fundamental Principles of Mathematical Sciences, 319 (Springer, Berlin, 1999).
[10]Choi, B. J., ‘Convergence of Mann’s alternating projections in CAT(𝜅) spaces’, Bull. Aust. Math. Soc. 98 (2018), 134143.
[11]Choi, B. J., Ji, U. C. and Lim, Y., ‘Convergences of alternating projections in CAT(𝜅) spaces’, Constr. Approx. 47 (2018), 391405.
[12]Choi, B. J., Ji, U. C. and Lim, Y., ‘Convex feasibility problems on uniformly convex metric spaces’, Optim. Methods Softw. 35 (2020), 2136.
[13]Combettes, P. L., ‘Hilbertian convex feasibility problem: convergence of projection methods’, Appl. Math. Optim. 35 (1997), 311330.
[14]Espínola, R. and Fernández-León, A., ‘CAT(𝜅)-spaces, weak convergence and fixed points’, J. Math. Anal. Appl. 353 (2009), 410427.
[15]He, J. S., Fang, D. H., López, G. and Li, C., ‘Mann’s algorithm for nonexpansive mappings in CAT(𝜅) spaces’, Nonlinear Anal. 75 (2012), 445452.
[16]Hundal, H. S., ‘An alternating projection that does not converge in norm’, Nonlinear Anal. 57 (2004), 3561.
[17]Kirk, W. A. and Panyanak, B., ‘A concept of convergence in geodesic spaces’, Nonlinear Anal. 68 (2008), 36893696.
[18]Kuwae, K., ‘Jensen’s inequality on convex spaces’, Calc. Var. Partial Differential Equations 49 (2014), 13591378.
[19]Lim, T. C., ‘Remarks on some fixed point theorems’, Proc. Amer. Math. Soc. 60 (1976), 179182.
[20]Ohta, S., ‘Convexities of metric spaces’, Geom. Dedicata 125 (2007), 225250.
[21]von Neumann, J., Functional Operators II: The Geometry of Orthogonal Spaces, Annals of Mathematics Studies, 22 (Princeton University Press, Princeton, NJ, 1950), reprint of Mimeographed Lecture Notes first distributed in 1933.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

$\unicode[STIX]{x1D6E5}$ -CONVERGENCES OF WEIGHTED AVERAGED PROJECTIONS IN $\text{CAT}(\unicode[STIX]{x1D705})$ SPACES

  • BYOUNG JIN CHOI (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.