scholarly journals A Modified Proximal Point Algorithm and Some Convergence Results

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Shengquan Weng ◽  
Dingping Wu ◽  
Zengfu Chao
Keyword(s):  
Nonexpansive Mapping ◽  
Convex Function ◽  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Numerical Example ◽  
Multivalued Nonexpansive Mapping ◽  
Convergence Results

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.

scholarly journals Proximal point algorithms involving Cesàro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4165-4176 ◽  
Author(s):  
Amna Kalsoom ◽  
Hafiz Fukhar-Ud-Din ◽  
Sara Najib
Keyword(s):  
Nonexpansive Mapping ◽  
Convex Function ◽  
Proximal Point Algorithm ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Proximal Point ◽  
Proximal Point Algorithms ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Total Asymptotically Nonexpansive Mappings

In this paper, we extend the proximal point algorithm proposed by Chang et al.[8] for total asymptotically nonexpansive mapping in CAT(0) spaces. We also demonstrate the ?-convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the Ces?ro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces.

scholarly journals Convergence Results for Proximal Point Algorithm in Complete Cat(0) Space for Multivalued Mappings

2021 ◽  
Vol 27 (1) ◽  
pp. 29-47
Author(s):  
Samir Dashputre ◽  
C. Padmavati ◽  
Kavita Sakure
Keyword(s):  
Proximal Point Algorithm ◽  
Nonexpansive Mappings ◽  
Multivalued Mappings ◽  
Proximal Point ◽  
Convergence Theorems ◽  
Numerical Example ◽  
Asymptotically Nonexpansive Mappings ◽  
Convergence Results ◽  
Asymptotically Nonexpansive

In this paper, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove some convergence theorems for the algorithm which was introduced by Shamshad Hussain et al. [18]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

scholarly journals On the proximal point algorithm and demimetric mappings in CAT(0) spaces

2018 ◽  
Vol 51 (1) ◽  
pp. 277-294 ◽  
Author(s):  
Kazeem O. Aremu ◽  
Chinedu Izuchukwu ◽  
Godwin C. Ugwunnadi ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Finite Family ◽  
Fixed Point Problems ◽  
Proximal Point ◽  
Numerical Example ◽  
Common Solution ◽  
Minimization Problems

Abstract In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space

2019 ◽  
Vol 10 (4) ◽  
pp. 437-446 ◽  
Author(s):  
Godwin C. Ugwunnadi ◽  
Chinedu Izuchukwu ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Metric Space ◽  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Uniformly Convex ◽  
Convex Metric Space ◽  
Proximal Point ◽  
Common Solution ◽  
Uniformly Convex Metric Space

AbstractWe prove some important properties of the p-resolvent mapping recently introduced by B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc. 31 2016, 4, 845–855, in p-uniformly convex metric space. Furthermore, we introduce a modified Mann-type PPA involving nonexpansive mapping and prove that the sequence generated by the algorithm converges to a common solution of a finite family of minimization problems, which is also a fixed point of a nonexpansive mapping in the framework of a complete p-uniformly convex metric space.

scholarly journals Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 347-360
Author(s):  
Mujahid Abbas ◽  
Hira Iqbal ◽  
Manuel de la Sen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Approximation Method ◽  
Multivalued Mappings ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Multivalued Nonexpansive Mapping ◽  
Convergence Results

AbstractWe prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete ℝ-trees without the endpoint condition.

scholarly journals Inexact Inertial Proximal Algorithm for Maximal Monotone Operators

2015 ◽  
Vol 23 (2) ◽  
pp. 133-146
Author(s):  
Hadi Khatibzadeh ◽  
Sajad Ranjbar
Keyword(s):  
Nonlinear Oscillator ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Algorithm ◽  
Weak And Strong Convergence ◽  
Proximal Point ◽  
Convergence Results ◽  
Inertial Proximal Algorithm

Abstract In this paper, convergence of the sequence generated by the inexact form of the inertial proximal algorithm is studied. This algorithm which is obtained by the discretization of a nonlinear oscillator with damping dynamical system, has been introduced by Alvarez and Attouch (2001) and Jules and Maingé (2002) for the approximation of a zero of a maximal monotone operator. We establish weak and strong convergence results for the inexact inertial proximal algorithm with and without the summability assumption on errors, under different conditions on parameters. Our theorems extend the results on the inertial proximal algorithm established by Alvarez and Attouch (2001) and rules and Maingé (2002) as well as the results on the standard proximal point algorithm established by Brézis and Lions (1978), Lions (1978), Djafari Rouhani and Khatibzadeh (2008) and Khatibzadeh (2012). We also answer questions of Alvarez and Attouch (2001).

scholarly journals A remark on recent results for finding zeroes of accretive operators

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Abdellatif Moudafi
Keyword(s):  
Proximal Point Algorithm ◽  
Accretive Operator ◽  
Accretive Operators ◽  
Proximal Point ◽  
Convergence Results ◽  
Yosida Approximate

By means of the Yosida approximate of an accretive operator, we extended two recent results by Chidume and Chidume and Zegeye (2003) to set-valued operators, and we made the connection with two recent convergence results obtained by Benavides etal for a relaxed version of the so-called proximal point algorithm.

scholarly journals Convergence Theorems of Modified Proximal Algorithms for Asymptotical Quasi-nonexpansive Mappings in CAT(0) Spaces

2018 ◽  
Vol 10 (2) ◽  
pp. 66
Author(s):  
Shengquan Weng ◽  
Dingping Wu
Keyword(s):  
Fixed Point ◽  
Convex Function ◽  
Finite Number ◽  
Iterative Process ◽  
Proximal Point Algorithm ◽  
Nonexpansive Mappings ◽  
Proximal Algorithms ◽  
Proximal Point ◽  
Convergence Theorems

In this paper, a new modified proximal point algorithm involving fixed point iterates of a finite number of asymptotically quasi-nonexpansive mappings in $CAT(0)$ spaces is proposed and been proved for the existence of a sequence generated by our iterative process converging to a minimizer of a convex function and a commen fixed point of a finite number of asymptotically quasi-nonexpansive mappings.

scholarly journals MODIFIED PROXIMAL POINT ALGORITHM FOR MINIMIZATION AND FIXED POINT PROBLEM IN CAT(0) SPACES

2021 ◽  
Vol 7 (1) ◽  
pp. 109
Author(s):  
Godwin Chidi Ugwunnadi
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Proximal Point Algorithm ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Proximal Point ◽  
Common Solution ◽  
Asymptotically Nonexpansive ◽  
Lower Semi

In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.

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