Article Sidebar

Submitted
January 24, 2020
Accepted
December 23, 2020
Published
March 31, 2021
Download
PDF
Statistic
Read Counter : 453 Download : 318

Downloads

Download data is not yet available.

Main Article Content

Abstract

In this paper, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove some convergence theorems for the algorithm which was introduced by Shamshad Hussain et al. [18]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

Keywords

CAT(0) spaces Nearly Lipschitzian mappings Proximal point algorithm Delta-convergence.

Article Details

How to Cite
Dashputre, S., Padmavati, C., & Sakure, K. (2021). Convergence Results for Proximal Point Algorithm in Complete Cat(0) Space for Multivalued Mappings. Journal of the Indonesian Mathematical Society, 27(1), 29–47. https://doi.org/10.22342/jims.27.1.899.29-47
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX
Crossref
Scopus
Google Scholar
Europe PMC

References

  1. Abbas M., Sahu D.R., Kadelburg Z., Fixed point theorems for Lipschitzian type mappings in
  2. CAT(0) spaces, Mathematical and Computer Modelling, 55 (2012) 1418{1427.
  3. Ambrosio L., Gigli N., Savare G. Gradient Flows in Metric Spaces and in the Space of Probability
  4. Measures, 2nd edn. Lectures in Mathematics ETH Zrich, Birkhuser, Basel (2008)
  5. Ansari, Q.H., Babu, F., Yao J., Regularization of proximal point algorithm in Hadamard mani-
  6. folds, Journal of Fixed Point Theory (2019) 21:25.
  7. Ansari, Q.H., Babu, F. Proximal point algorithm for inclusion problems in Hadamard manifolds
  8. with applications , Optim. Lett. (2019) 1:21.
  9. Ahmadi P., Khatibzadeh H., On the Convergence of Inexact Proximal Point Algorithm on
  10. Hadamard Manifolds, Taiwanese J. Math. 18 (2014), 419-433.
  11. Ariza-Ruiz D., Leustean L., Lopez G., Firmly nonexpansive mappings in classes of geodesic
  12. spaces, Trans. Am. Math. Soc. 366, 4299􀀀4322 (2014)
  13. Bacak, M., The proximal point algorithm in metric spaces, Isr. J. Math.194, 689􀀀701 (2013).
  14. Bacak M., Reich S., The asymptotic behavior of a class of nonlinear semigroups in Hadamard
  15. spaces, J. Fixed Point Theory Appl. 16, 189􀀀202 (2014).
  16. Bento G.C., CruzNeto J.X., Oliveira P.R., A New Approach to the Proximal Point Method:
  17. Convergence on General Riemannian Manifolds, J. Optim. Theory Appl. 168 (2016), 743-755.
  18. Bento G.C., Ferreira O.P., Pereira Y.R.L, Proximal point method for vector optimization on
  19. Hadamard manifold. Operation Research Letters 46(1) (2018) 13:18.
  20. Boikanyo O.A., Morosanu G., A proximal point algorithm converging strongly for general errors,
  21. Optim. Lett. 4, 635􀀀641 (2010).
  22. Bruck R.E., Reich S., Nonexpansive projections and resolvents of accretive operators in Banach
  23. spaces, Houston J. Math. 3, 459􀀀470 (1977).
  24. Chang S.S., Yao J.C., Wang L., Qin L.J., Some Convergence Theorems Involving Proximal Point
  25. and Common Fixed Points for Asymptotically Nonexpansive Mappings in CAT(0) Spaces, Fixed
  26. Point Theory Appl. 2016, 68 : 2016.
  27. Cholamjiak P., Abdou A.A., Cho Y.J., Proximal point algorithms involving xed points of non-
  28. expansive mappings in CAT(0) spaces, Fixed Point Theory Appl.227, 1􀀀13 (2015).
  29. Cholamjiak P., The Modied Proximal Point Algorithm in CAT(0) Space, Optim. Lett. 9,
  30. {1410 (2015).
  31. Ferreira O.P., Oliveira P.R., Proximal Point Algorithm on Riemannian Manifolds, Optimization
  32. (2002), 257-270.
  33. Guler, O., On the convergence of the proximal point algorithm for convex minimization, SIAM
  34. J. Control Optim. 29 , 403􀀀419 (1991).
  35. Hussain S., Singh N., 􀀀 convergence for Proximal Point Algorithm and Fixed Point Problem
  36. in CAT(0) Space, Fixed Point Theory Appl. 2019, 8 : 2019.
  37. Jost J., Convex functionals and generalized harmonic maps into spaces of nonpositive curvature,
  38. Comment. Math. Helv.70, 659673 (1995).
  39. Kamimura S., Takahashi W., Approximating solutions of maximal monotone operators in Hilbert
  40. spaces, J. Approx. Theory 106, 226􀀀240 (2000).
  41. Kim J.K., Dashputre S., Das A.K., Convergence theorems of S-iteration process for Lipschitzian
  42. type multi-valued mappings in Banach Spaces, Global Journal of Pure and Applied Mathematics,
  43. (1) (2017), 121-135.
  44. Li C., Lopez G., Martin-Marquez V., Monotone Vector Fields and the Proximal Point Algorithm
  45. on Hadamard Manifolds, J. London Math. Soc. 79 (2009), 663-683.
  46. Markin J.T., Continuous Dependence of Fixed Point Sets, Proc. Am. Math. Soc. 38, 545{547
  47. (1973).
  48. Martinet, B., Regularisation dinquations variationnelles par approximations successives, Rev.
  49. Fr. Inf. Rech. Oper. 4 , 154􀀀158 (1970).
  50. Mayer U.F.,Gradient
  51. ows on nonpositively curved metric spaces and harmonic maps, Commun.
  52. Anal. Geom.6, 199253 (1998).
  53. Nadler S.B., Multivalued Contraction Mappings, Pac. J. Math. 30, 475{488 (1969).
  54. Pakkaranang, N., Kumam, P., Cho, Y.J., Proximal point algorithms for solving convex minimiza-
  55. tion problem and common xed points problem of asymptotically quasi-nonexpansive mappings
  56. in CAT(0) spaces with convergence analysis, Numer. Algorithms, 78(3) (2018), 827{845.
  57. Phuengrattana W., Onjai-uea N., Cholamjiak P., Modied Proximal Point Algorithms for Solving
  58. Constrained Minimization and Fixed Point Problems in Complete CAT(0) Spaces, Meediterr. J.
  59. Math. (2018) 15:97.
  60. Reich S., Saback S., Two strong convergence theorems for a proximal method in re
  61. exive Banach
  62. spaces, Numer. Funct. Anal. Optim.31, 22􀀀44 (2010).
  63. Reich S., Salinas Z., Weak convergence of innite products of operators in Hadamard spaces,
  64. Rend. Circ. Mat. Palermo 65, 55􀀀71 (2016).
  65. Rockafeller R.T. Monotone operators and the proximal point algorithm, SIAM J. Control Optim.
  66. , 877􀀀898 (1976).
  67. Shimizu T., Takahashi W., Fixed Points of Multivalued Mappings in Certain Convex Metric
  68. Space, Topol. Methods Nonlinear Analysis 8, 197{203 (1996).
  69. Suparatulatorn R., Cholamjiak P., Suantai S., On solving the minimization problem and the
  70. xed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods Softw. 32,
  71. (2017).
  72. Wang, J., Li, C., Lopez, G., Yao J., Convergence analysis of inexact proximal point algorithms
  73. on Hadamard manifolds . J. Glob. Optim. (2015) 61: 553.