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Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces

Year 2018, Volume: 1 Issue: 2, 49 - 57, 31.08.2018

Shengquan Weng Dingping Wu

Abstract

In this paper, a new modified proximal point algorithm is proposed for finding a common element of the set of fixed points of a single-valued nonexpansive mapping, and the set of fixed points of a multivalued nonexpansive mapping, and the set of minimizers of convex and lower semicontinuous functions. We obtain convergence of the proposed algorithm to a common element of three sets in CAT(0) spaces.

Keywords

CAT(0) space proximal point algorithm fixed point resolvent identity functional analysis

References

  • [1] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. Inst. Hautes Etudes Sci. Publ. Math. 41(1972), 5-251.
  • [2] S. Dhompongsa, B. Panyanak, On ∆−convergence theorems in CAT(0) spaces.Comput. Math. Appl.56(2008), 2572-2579.
  • [3] M. Ba˘ cák, The proximal point algorithm in metric spaces. Isr. J. Math. 194(2013),689-701.
  • [4] O. Guler, On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim.29(1991), 403-419.
  • [5] D. Ariza-Ruiz, L. Leustean, G. Lopez, Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc.366 (2014), 4299-4322.
  • [6] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70(1995), 659-673.
  • [7] S. Suantai, W. Phuengrattana, Proximal Point Algorithms for a Hybrid Pair of Nonexpansive Single-Valued and MultiValued Mappings in Geodesic Metric Spaces. (2017).
  • [8] T. Rockafellar, R.J. Wets, Variational Analysis. Springer, Berlin(2005)
  • [9] S. Dhompongsa, W.A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal.65 (2006), 762-772.
  • [10] W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces. Nonlinear Anal. 68 (2008), 3689-3696.
  • [11] S. Dhompongsa, W.A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal.8(2007), 35-45.
  • [12] L. Ambrosio, N. Gigli, G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zrich, 2nd edn. Birkhuser, Basel (2008).

Year 2018, Volume: 1 Issue: 2, 49 - 57, 31.08.2018

Shengquan Weng Dingping Wu

Abstract

References

  • [1] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. Inst. Hautes Etudes Sci. Publ. Math. 41(1972), 5-251.
  • [2] S. Dhompongsa, B. Panyanak, On ∆−convergence theorems in CAT(0) spaces.Comput. Math. Appl.56(2008), 2572-2579.
  • [3] M. Ba˘ cák, The proximal point algorithm in metric spaces. Isr. J. Math. 194(2013),689-701.
  • [4] O. Guler, On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim.29(1991), 403-419.
  • [5] D. Ariza-Ruiz, L. Leustean, G. Lopez, Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc.366 (2014), 4299-4322.
  • [6] J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70(1995), 659-673.
  • [7] S. Suantai, W. Phuengrattana, Proximal Point Algorithms for a Hybrid Pair of Nonexpansive Single-Valued and MultiValued Mappings in Geodesic Metric Spaces. (2017).
  • [8] T. Rockafellar, R.J. Wets, Variational Analysis. Springer, Berlin(2005)
  • [9] S. Dhompongsa, W.A. Kirk, B. Sims, Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal.65 (2006), 762-772.
  • [10] W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces. Nonlinear Anal. 68 (2008), 3689-3696.
  • [11] S. Dhompongsa, W.A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal.8(2007), 35-45.
  • [12] L. Ambrosio, N. Gigli, G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zrich, 2nd edn. Birkhuser, Basel (2008).
There are 12 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Shengquan Weng This is me

Dingping Wu This is me

Publication Date August 31, 2018
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Weng, S., & Wu, D. (2018). Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces. Results in Nonlinear Analysis, 1(2), 49-57.
AMA Weng S, Wu D. Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces. RNA. August 2018;1(2):49-57.
Chicago Weng, Shengquan, and Dingping Wu. “Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces”. Results in Nonlinear Analysis 1, no. 2 (August 2018): 49-57.
EndNote Weng S, Wu D (August 1, 2018) Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces. Results in Nonlinear Analysis 1 2 49–57.
IEEE S. Weng and D. Wu, “Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces”, RNA, vol. 1, no. 2, pp. 49–57, 2018.
ISNAD Weng, Shengquan - Wu, Dingping. “Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces”. Results in Nonlinear Analysis 1/2 (August 2018), 49-57.
JAMA Weng S, Wu D. Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces. RNA. 2018;1:49–57.
MLA Weng, Shengquan and Dingping Wu. “Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces”. Results in Nonlinear Analysis, vol. 1, no. 2, 2018, pp. 49-57.
Vancouver Weng S, Wu D. Some Convergence Theorems of Modified Proximal Point Algorithms for Nonexpansive Mappings in CAT(0) Spaces. RNA. 2018;1(2):49-57.
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