Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

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Convergence theorems for generalized projections and maximal monotone operators in Banach spaces

Abstract and Applied Analysis ◽

10.1155/s1085337503207065 ◽

2003 ◽

Vol 2003 (10) ◽

pp. 621-629 ◽

Cited By ~ 17

Author(s):

Takanori Ibaraki ◽

Yasunori Kimura ◽

Wataru Takahashi

Keyword(s):

Banach Space ◽

Pointwise Convergence ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Convex Banach Space ◽

Limit Set ◽

Strong And Weak Convergence ◽

Strictly Convex Banach Space ◽

Generalized Projections

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

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A weak convergence theorem for relatively nonexpansive mappings and maximal monotone operators in a Banach space

Journal of Applied and Numerical Optimization ◽

10.23952/jano.3.2021.1.11 ◽

2021 ◽

Vol 3 (1) ◽

Keyword(s):

Banach Space ◽

Weak Convergence ◽

Convergence Theorem ◽

Nonexpansive Mappings ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Relatively Nonexpansive Mappings ◽

Relatively Nonexpansive ◽

Weak Convergence Theorem

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Strong convergence of an iterative sequence for maximal monotone operators in a Banach space

Abstract and Applied Analysis ◽

10.1155/s1085337504309036 ◽

2004 ◽

Vol 2004 (3) ◽

pp. 239-249 ◽

Cited By ~ 84

Author(s):

Fumiaki Kohsaka ◽

Wataru Takahashi

Keyword(s):

Banach Space ◽

Strong Convergence ◽

Variational Inequality Problem ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Iterative Sequence ◽

Strong Convergence Theorem ◽

Convex Minimization Problem ◽

Proximal Point

We first introduce a modified proximal point algorithm for maximal monotone operators in a Banach space. Next, we obtain a strong convergence theorem for resolvents of maximal monotone operators in a Banach space which generalizes the previous result by Kamimura and Takahashi in a Hilbert space. Using this result, we deal with the convex minimization problem and the variational inequality problem in a Banach space.

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Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

Optimization ◽

10.1080/02331934.2017.1343330 ◽

2017 ◽

Vol 66 (10) ◽

pp. 1623-1646

Author(s):

Wongvisarut Khuangsatung ◽

Atid Kangtunyakarn

Keyword(s):

Minimization Problem ◽

Maximal Monotone ◽

Iterative Algorithms ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Convex Minimization ◽

Convex Minimization Problem ◽

Constrained Convex Minimization ◽

Constrained Convex Minimization Problem

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Strong and Weak Convergence Theorems for Common Solutions of Generalized Equilibrium Problems and Zeros of Maximal Monotone Operators

Fixed Point Theory and Applications ◽

10.1155/2010/590278 ◽

2010 ◽

Vol 2010 ◽

pp. 1-34 ◽

Cited By ~ 5

Author(s):

L.-C. Zeng ◽

Q. H. Ansari ◽

David S. Shyu ◽

J.-C. Yao

Keyword(s):

Weak Convergence ◽

Equilibrium Problems ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Monotone Operators ◽

Convergence Theorems ◽

Strong And Weak Convergence ◽

Generalized Equilibrium

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Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/174796 ◽

2011 ◽

Vol 2011 ◽

pp. 1-31 ◽

Cited By ~ 3

Author(s):

Kriengsak Wattanawitoon ◽

Poom Kumam

Keyword(s):

Equilibrium Problems ◽

Maximal Monotone ◽

Maximal Monotone Operators ◽

Zero Point ◽

Monotone Operators ◽

Common Element ◽

Proximal Point ◽

Strong And Weak Convergence ◽

Mixed Equilibrium Problems ◽

Inverse Strongly Monotone

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors.

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

Journal of Applied Mathematics ◽

10.1155/2014/414031 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 2

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.

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