scholarly journals The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Oganeditse Aaron Boikanyo
Keyword(s):  
Real Hilbert Space ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Viscosity Approximation ◽  
Convergence Results ◽  
Bounded Below ◽  
Inexact Algorithm ◽  
Cocoercive Operator

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operatorAand maximal monotone operatorsBwithD(B)⊂H:xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, forn=1,2,…,for givenx1in a real Hilbert spaceH, where(αn),(γn), and(δn)are sequences in(0,1)withαn+γn+δn=1for alln≥1,(en)denotes the error sequence, andf:H→His a contraction. The algorithm is known to converge under the following assumptions onδnanden: (i)(δn)is bounded below away from 0 and above away from 1 and (ii)(en)is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i)(δn)is bounded below away from 0 and above away from 3/2 and (ii)(en)is square summable in norm; and we still obtain strong convergence results.

scholarly journals Quantitative results on Fejér monotone sequences

2017 ◽  
Vol 20 (02) ◽  
pp. 1750015 ◽  
Author(s):  
Ulrich Kohlenbach ◽  
Laurenţiu Leuştean ◽  
Adriana Nicolae
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Point ◽  
Convergence Results ◽  
Asymptotically Nonexpansive ◽  
Firmly Nonexpansive ◽  
Quantitative Results ◽  
Iterative Procedures ◽  
Fixed Point Iterations

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejér monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations [Formula: see text] for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called [Formula: see text]-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.

scholarly journals On an Iterative Method for Finding a Zero to the Sum of Two Maximal Monotone Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hongwei Jiao ◽  
Fenghui Wang
Keyword(s):  
Weak Convergence ◽  
Iterative Method ◽  
Variational Inequalities ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Gradient Projection ◽  
Projection Algorithms

In this paper we consider a problem that consists of finding a zero to the sum of two monotone operators. One method for solving such a problem is the forward-backward splitting method. We present some new conditions that guarantee the weak convergence of the forward-backward method. Applications of these results, including variational inequalities and gradient projection algorithms, are also considered.

scholarly journals Differential game with switching controls on Hilbert space

1997 ◽  
Vol 39 (2) ◽  
pp. 230-256
Author(s):  
Siu Pang Yung
Keyword(s):  
Hilbert Space ◽  
Differential Game ◽  
Viscosity Solution ◽  
Real Hilbert Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Evolution System ◽  
Value Of The Game ◽  
Existence Of Value

AbstractWe study differential game problems in which the players can select different maximal monotone operators for the governing evolution system. Setting up our problem on a real Hilbert space, we show that the Elliott-Kalton upper and lower value of the game are viscosity solution of some Hamilton-Jacobi-Isaacs equations. Uniqueness is obtained by assuming condition analogous to the classical Isaacs condition, and thus the existence of value of the game follows.

scholarly journals Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 773 ◽  
Author(s):  
Mihai Postolache ◽  
Ashish Nandal ◽  
Renu Chugh
Keyword(s):  
Fixed Point ◽  
Splitting Method ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Feasibility Problem ◽  
Inclusion Problem ◽  
Special Cases ◽  
Implicit Rule

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

scholarly journals A strong convergence theorem for maximal monotone operators in Banach spaces with applications

2020 ◽  
Vol 36 (2) ◽  
pp. 229-240
Author(s):  
C. E. CHIDUME ◽  
◽  
G. S. DE SOUZA ◽  
O. M. ROMANUS ◽  
U. V. NNYABA ◽  
...  
Keyword(s):  
Banach Space ◽  
Real Hilbert Space ◽  
Maximal Monotone ◽  
General Setting ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Strong Convergence Theorem ◽  
Proximal Point ◽  
Monotone Map ◽  
Maximal Monotone Map

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex anduniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zeroof the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem com-plements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergencetheorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally,numerical experiments are presented to illustrate the convergence of our algorithm.

scholarly journals Approximation of zeros of m-accretive mappings, with applications to Hammerstein integral equations

2020 ◽  
Vol 36 (1) ◽  
pp. 59-69
Author(s):  
CHARLES CHIDUME ◽  
GERALDO SOARES De SOUZA ◽  
VICTORIA UKAMAKA NNYABA
Keyword(s):  
Real Hilbert Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Accretive Operator ◽  
Hammerstein Integral Equation ◽  
Proximal Point ◽  
Hammerstein Integral Equations ◽  
Uniformly Smooth ◽  
Accretive Mappings

"An algorithm for approximating zeros of m-accretive operators is constructed in a uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a zero of an m-accretive operator. In the case of a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar for approximating zeros of maximal monotone operators. Furthermore, the convergence theorem proved is applied to approximate a solution of a Hammerstein integral equation. Finally, numerical experiments are presented to illustrate the convergence of our algorithm."

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).

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