A modified viscosity implicit-type proximal point algorithm for monotone inclusions and asymptotically nonexpansive mappings in Hadamard spaces

The purpose of this article is to propose a modified viscosity implicit-type proximal point algorithm for approximating a common solution of a monotone inclusion problem and a fixed point problem for an asymptotically nonexpansive mapping in Hadamard spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved. Our results extend and complement some recent results in this direction.

[1]  B. Martinet,et al.  R'egularisation d''in'equations variationnelles par approximations successives , 1970 .

[2]  Prasit Cholamjiak,et al.  The modified proximal point algorithm in CAT(0) spaces , 2015, Optim. Lett..

[3]  Heinz H. Bauschke,et al.  Projection and proximal point methods: convergence results and counterexamples , 2004 .

[4]  M. Bacák.  The proximal point algorithm in metric spaces , 2012, 1206.7074.

[5]  溫慶豐,et al.  Proximal point algorithms involving Cesaro type mean of asymptotically nonexpansive mappings in CAT(0) spaces , 2016 .

[6]  H. Khatibzadeh,et al.  New results on the proximal point algorithm in non-positive curvature metric spaces , 2017 .

[7]  Yeol Je Cho,et al.  Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0)$\operatorname{CAT}(0)$ spaces , 2015 .

[8]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

[9]  Paul-Emile Maingé,et al.  Viscosity methods for zeroes of accretive operators , 2006, J. Approx. Theory.

[10]  Chong Li,et al.  Monotone vector fields and the proximal point algorithm on Hadamard manifolds , 2009 .

[11]  H. Khatibzadeh,et al.  MONOTONE OPERATORS AND THE PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) METRIC SPACES , 2016, Journal of the Australian Mathematical Society.

[12]  Jacques Tits,et al.  Groupes Réductifs Sur Un Corps Local , 1972 .

[13]  H. Khatibzadeh,et al.  Strong and $$\Delta $$Δ-Convergence to a Zero of a Monotone Operator in CAT(0) Spaces , 2017 .

[14]  Bijan Ahmadi Kakavandi,et al.  Weak topologies in complete $CAT(0)$ metric spaces , 2012 .

[15]  I. D. Berg,et al.  Quasilinearization and curvature of Aleksandrov spaces , 2008 .

[16]  Bruna Tanaka Cremonini,et al.  Buildings , 1995, Data, Statistics, and Useful Numbers for Environmental Sustainability.

[17]  William A. Kirk,et al.  Fixed points of uniformly lipschitzian mappings , 2006 .

[18]  Suthep Suantai,et al.  On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces , 2017, Optim. Methods Softw..

[19]  Wataru Takahashi,et al.  Viscosity approximation methods for resolvents of accretive operators in Banach spaces , 2007 .

[20]  Jacques Tits,et al.  Groupes réductifs sur un corps local , 1972 .

[21]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[22]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[23]  Simeon Reich,et al.  The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces , 2014, 1405.6637.

[24]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[25]  Jen-Chih Yao,et al.  Some convergence theorems involving proximal point and common fixed points for asymptotically nonexpansive mappings in CAT(0)$\operatorname {CAT}(0)$ spaces , 2016 .

[26]  Li Yang,et al.  Demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces , 2012, Appl. Math. Comput..

[27]  X. Qin,et al.  Weak convergence of a splitting algorithm in Hilbert spaces , 2017 .

[28]  Massoud Amini,et al.  Duality and subdifferential for convex functions on complete CAT(0) metric spaces , 2010 .

[29]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[30]  Jen-Chih Yao,et al.  Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces , 2018 .

[31]  William A. Kirk,et al.  A concept of convergence in geodesic spaces , 2008 .