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

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
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.

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 An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lingling Huang ◽  
Sanyang Liu ◽  
Weifeng Gao
Keyword(s):  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Computational Results ◽  
Monotone Inclusion ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

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