Viscosity approximation method for monotone operators in Hadamard spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Sciences, Yasouj University, Yasouj, Iran

2 Department of Mathematics, College of Sciences, Higher Education Center of Eghlid, Eghlid, Iran

Abstract

In this article, we suggest and analyze a  viscosity approximation method to a zero of a monotone operator in the setting of Hadamard spaces. We derive the convergence of sequences generated by the proposed viscosity methods under some suitable assumptions. Also, some applications to solve the variational inequality, optimization and fixed point problems are given on Hadamard spaces.

Keywords

[1] B. Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141 (2013),
1029–1039.
[2] B. Ahmadi Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric
spaces, Nonlinear Anal. 73 (2010), 3450–3455.
[3] M. Baˇc´ak, The proximal point algorithm in metric spaces, Isr. J. Math. 194 (2013), 689-701.
[4] M. Baˇc´ak, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and
Applications, 22. De Gruyter, Berlin, 2014.
[5] I.D. Berg and I.G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata 133 (2008),
195-218.
[6] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Vol. 319 SpringerVerlag, Berlin, Heidelberg, New York, 1999.
[7] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate studies in Math., Vol. 33, Amer.
Math. Soc., Providence, RI, 2001.
[8] H. Dehghan and J. Rooin,A characterization of metric projection in CAT(0) spaces, Int. Conf. Funct. Equ. Geo.
Funct. Appl. Payame Noor University, Tabriz, 2012, pp. 41–43.
[9] M. Gromov and S.M. Bates, Metric structures for Riemannian and Non-Riemanian Spaces, Vol. 152, Boston:
Birkh¨auser, 1999.
[10] M.T. Heydari and S. Ranjbar, Halpern-type proximal point algorithm in complete CAT(0) metric spaces , An.
Stiint. Univ. Ovidius Constanta Ser. Mat. 24 (2016), 141–159.
[11] J. Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Mathematics, ETH Zurich. Basel:
Birkh¨auser, 1997.
[12] S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J.
App. Theory 106 (2000), 226–240.
[13] H. Khatibzadeh and S. Ranjbar,On the Hapern iteration in CAT(0) spaces, Ann. Funct. Anal. 6 (2015), 155165.
[14] H. Khatibzadeh and S. Ranjbar, A variational inequality in complete CAT(0) spaces, J. Fixed Point Theory Appl.
17 (2015), 557–574.
[15] H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric
spaces, J. Aust. Math. Soc. 103 (2017), no. 1, 70–90.
[16] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689–3696.
[17] C. Li, G. L´opez and V. Mart´ın-M´arquez, Monotone vector fields and the proximal point algorithm, J. London
Math. Soc. 679 (2009), 663–683.
[18] T.C. Lim, Remarks on some fixed point theorems, Proc. Am. Math. Soc. 60 (1976), 179–182.
[19] B. Martinet, R´egularistion d’in´equations variationnelles par approximations successive, Rev. Fran¸caise Inft.
Recher op´erationnelle 4 (1970), 154–158.
[20] J.X. Da Cruz Neto, O.P. Ferreira, L.R. Lucambio Prez and S.Z. Nemeth, Convex and monotone transformable
mathematical programming problems and a proximal-like point method, J. Global Optim. 35 (2006), no. 1, 53-–69.[21] S. Ranjbar, W-convergence of the proximal point algorithm in complete CAT(0) metric spaces, Bull. Iran. Math.
Soc. 43 (2017), 817–834.
[22] S. Ranjbar and H. Khatibzadeh, Strong and ∆-convergence to a zero of a monotone operator in CAT(0) spaces,
Mediterr. J. Math. 14 (2017), no. 2, 1–15.
[23] T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 79–83.
[24] W. Takahashi, Viscosity approximation methods for resolvents of accretive operators in Banach spaces, J. Fixed
Point Theory Appl. 1 (2007), 135–147.
[25] S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces,
Nonlinear Anal. 75 (2012), 742–750.
Volume 13, Issue 2
July 2022
Pages 1023-1032
  • Receive Date: 18 September 2020
  • Revise Date: 05 August 2021
  • Accept Date: 22 August 2021