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).