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.

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 A New Approximation Method for Finding Zeros of Maximal Monotone Operators

2021 ◽  
Vol 31 (2) ◽  
pp. 117-124
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Proximal Point ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

scholarly journals A Modified Regularization Method for the Proximal Point Algorithm

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Shuang Wang
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Regularization Method ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Proximal Point ◽  
Weaker Conditions

Under some weaker conditions, we prove the strong convergence of the sequence generated by a modified regularization method of finding a zero for a maximal monotone operator in a Hilbert space. In addition, an example is also given in order to illustrate the effectiveness of our generalizations. The results presented in this paper can be viewed as the improvement, supplement, and extension of the corresponding results.

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