Fixed Point of Lipschitz Quasi Nonexpansive Mappings

Authors: M. Moosavi, H. Khatibzadeh

Abstract:

In this article, we study demiclosed and strongly quasi-nonexpansive of a sequence generated by the proximal point algorithm for a finite family of quasi-nonexpansive mappings in Hadamard spaces. Δ-convergence of iterations for the sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them to a common fixed point of sequence are also established. Our results generalize and improve several previously known results of the existing literature.

Keywords: Fixed point, Hadamard space, proximal point algorithm, quasi-nonexpansive sequence of mappings, resolvent.

References:

[1] P. Ahmadi and H. Khatibzadeh, “On the convergence of inexact proximal point algorithm on Hadamard manifolds,” Taiwanese J. Math. 2014, pp. 419–433.
[2] D. Ariza-Ruiz, L. Leustean, and G. Lopez-Acedo, “Firmly nonexpansive mappings in classes of geodesic spaces,” Trans. Amer. Math. Soc. 2014, pp. 4299–4322.
[3] M. Bacak, “The proximal point algorithm in metric spaces,” Israel J. Math. 2013, pp. 689–701.
[4] M. Bacak, I. Searston, and B. Sims, “Alternating projections in CAT(0) spaces,” J. Math. Anal. Appl. 2012, pp. 599-607.
[5] M. Bacak, S. Reich, “The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces,” J. Fixed Point Theory Appl. 2014, pp. 189–202.
[6] H.H. Bauschke, P.L. Combettes, and S. Reich, “The asymptotic behavior of the composition of two resolvents,” Nonlinear Anal. 2005, pp. 283-301.
[7] G. C. Bento, O. P. Ferreira, and P. R. Oliveira, “Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds,” Nonlinear Anal. 2010, pp. 564–572.
[8] O.A. Boikanyo and G. Morosanu, “On the method of alternating resolvents,” Nonlinear Anal. 2011, pp. 5147-5160.
[9] F. Bruhat and J. Tits, “Groupes r´eductifs sur un corps local: I. donnees radicielles valu´ees,” Publications Math´ematiques de l’IHES´, vol. 41, no. 1, 1972.
[10] S. Dhompongsa, W. A. Kirk and B. Sims, “Fixed points of uniformly Lipschitzian mappings,” Nonlinear Anal. 2006, pp. 762–772.
[11] O. P. Ferreira and P. R. Oliveira, “Proximal point algorithm on Riemannian manifolds,” Optim. 2002, pp. 257–270.
[12] W. A. Kirk, and B. Panyanak, “A concept of convergence in geodesic spaces,” Nonlinear Anal. 2008, pp. 3689–3696.
[13] H. Khatibzadeh and V. Mohebbi, “On the Iterations of a Sequence of Strongly Quasi-Nonexpansive Mappings with Applications,” Numer. Funct. Anal. Optim. 2020, pp. 231-256.
[14] C. Li, G. Lopez, and V. Martin-Marquez, “Monotone vector fields and the proximal point algorithm on Hadamard manifolds,” J. London Math. Soc. 2009, pp. 663–683.
[15] B. Martinet, “R’egularisation dinequations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle,” 1970, pp. 154-158.
[16] E. A. Papa Quiroz and P. R. Oliveira, “Proximal point method for minimizing quasi-convex locally Lipschitz functions on Hadamard manifolds” Nonlinear Anal. 2012, pp. 5924–5932.
[17] R.T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific J. Math. 1970, pp. 209–216.
[18] G. Tang and Y. Xiao, “A note on the proximal point algorithm for pseudo-monotone variational inequalities on Hadamard d manifolds,” Adv. Nonlinear Var. Inequal. 2015, pp. 58-69.
[19] G. Tang, L. Zhou, and N. Huang’ “The proximal point algorithm for pseudo-monotone variational inequalities on Hadamard manifolds,” Optim. Lett. 2013, pp. 779–790.