Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

2010 ◽  
Vol 31 (9) ◽  
pp. 1053-1071 ◽  
Author(s):  
Javad Mashreghi ◽  
Mostafa Nasri
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Proximal Point Algorithm ◽  
Equilibrium Problems ◽  
Proximal Point ◽  
Inexact Proximal

scholarly journals Iterative methods for solving split feasibility problems and fixed point problems in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5345-5353
Author(s):  
Min Liu ◽  
Shih-Sen Changb ◽  
Ping Zuo ◽  
Xiaorong Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Recent Result ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Proximal Point ◽  
Split Feasibility Problems ◽  
Split Feasibility

In this paper, we consider a class of split feasibility problems in Banach space. By using shrinking projective method and the modified proximal point algorithm, we propose an iterative algorithm. Under suitable conditions some strong convergence theorems are proved. Our results extend a recent result of Takahashi-Xu-Yao (Set-Valued Var. Anal. 23, 205-221 (2015)) from Hilbert spaces to Banach spaces. Moreover, the method of proof is also different.

scholarly journals Strong Convergence Theorems for Zeros of Bounded Maximal Monotone Nonlinear Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
C. E. Chidume ◽  
N. Djitté
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Iteration Process ◽  
Accretive Operator ◽  
Initial Vector ◽  
Proximal Point ◽  
Nonlinear Operators ◽  
Uniformly Continuous

An iteration process studied by Chidume and Zegeye 2002 is proved to convergestronglyto a solution of the equationAu=0whereAis a boundedm-accretive operator on certain real Banach spacesEthat includeLpspaces2≤p<∞.The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets ofE, setbacks associated with the classicalproximal point algorithmof Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.

MONOTONE OPERATORS AND THE PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) METRIC SPACES

2016 ◽  
Vol 103 (1) ◽  
pp. 70-90 ◽  
Author(s):  
HADI KHATIBZADEH ◽  
SAJAD RANJBAR
Keyword(s):  
Strong Convergence ◽  
Monotone Operator ◽  
Proximal Point Algorithm ◽  
Metric Spaces ◽  
Monotone Operators ◽  
Proximal Point ◽  
Range Condition ◽  
Special Cases ◽  
Inexact Proximal

In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

scholarly journals A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mei Yuan ◽  
Xi Li ◽  
Xue-song Li ◽  
John J. Liu
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Projection Method ◽  
Convergence Theorem ◽  
Projection Operator ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalizedf-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

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