Geometric Convergence of Iterative Methods for a Problem with M-matrices and Diagonal Multivalued Operators

2002 ◽  
Vol 2 (1) ◽  
pp. 26-40 ◽  
Author(s):  
Alexander Lapin
Keyword(s):  
Iterative Methods ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Mesh Approximation ◽  
Finite Dimensional ◽  
Multivalued Operators ◽  
Geometric Convergence ◽  
Parallel Iterative ◽  
A Free Boundary Problem

Abstract A finite-dimensional problem with several M-matrices and diagonal maximal monotone operators is studied. This problem includes variational inequalities with M-matrices as a partial case and appears, in particular, as a mesh approximation for a free boundary problem with several constraints and nonlinear relations. The existence of an unique solution for the problem is studied, as well as the convergence and geometric rate of the convergence for a class of the iterative methods, the Schwarz alternating-type methods among them. The application of the general results to a mesh scheme for a dam problem is considered. Parallel iterative methods are constructed on the basis of the domain decomposition, geometric convergence of these methods is justified.

Solution of a Finite-Dimensional Problem With M-mappings and Diagonal Multivalued Operators

2001 ◽  
Vol 1 (3) ◽  
pp. 242-264 ◽  
Author(s):  
E. Laitinen ◽  
Alexander Lapin
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Dimensional Problem ◽  
Existence Of A Solution ◽  
Finite Dimensional ◽  
Monotone Dependence ◽  
Multivalued Operators ◽  
The Right ◽  
Uniqueness Of A Solution

AbstractThe existence of a solution of the finite-dimensional problem with continuous M-mappings and multivalued diagonal maximal monotone operators on an ordered interval, which is formed by the so-called subsolution, is proved. Under several additional assumptions on the operators the monotone dependence of a solution upon the right-hand side is investigated. This result implies, in particular, the uniqueness of a solution and serves as a basis for the analysis of the convergence for a multisplitting iterative method. As an illustrative example, the finite difference scheme approximating a model variational inequality is studied by using the general results.

scholarly journals On the maximal monotonicity and the range of the sum of nonlinear maximal monotone operators

1991 ◽  
Vol 34 (1) ◽  
pp. 143-153 ◽  
Author(s):  
J. R. L. Webb ◽  
Weiyu Zhao
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Maximal Monotonicity ◽  
Multivalued Operators ◽  
Nonlinear Maximal Monotone ◽  
Nondecreasing Functions

Conditions are given on two maximal monotone (multivalued) operators A and B which ensure that A + B is also maximal. One condition used is that ∥Bx∥≦k(∥x∥)Ax| +d|(A + B)x| + c(∥x∥) for every x∈D(A)⊆D(B), where 0≦k(r)<1, and c(r)≧0 are nondecreasing functions, and 0≦d≦1 is a constant. Here, for a set C, |C| denotes inf{∥y∥:y∈C}. This extends the well known result which has d = 0 (and is used in the proof here). The second part of the paper uses similar hypotheses to give conditions under which the range of the sum, R(A + B), has the same interior and same closure as the sum of the ranges, R(A) + R(B).

scholarly journals Topological degree and application to a parabolic variational inequality problem

2001 ◽  
Vol 25 (4) ◽  
pp. 273-287 ◽  
Author(s):  
A. Addou ◽  
B. Mermri
Keyword(s):  
Variational Inequality ◽  
Topological Degree ◽  
Variational Inequality Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parabolic Variational Inequality ◽  
Inequality Problem ◽  
Monotone Map ◽  
New Formulation

We are interested in constructing a topological degree for operators of the formF=L+A+S, whereLis a linear densely defined maximal monotone map,Ais a bounded maximal monotone operators, andSis a bounded demicontinuous map of class(S+)with respect to the domain ofL. By means of this topological degree we prove an existence result that will be applied to give a new formulation of a parabolic variational inequality problem.

scholarly journals Theoretical Study of a Chain Sliding on a Fixed Support

2009 ◽  
Vol 2009 ◽  
pp. 1-19 ◽  
Author(s):  
Jérôme Bastien ◽  
Claude-Henri Lamarque
Keyword(s):  
Dry Friction ◽  
Theoretical Study ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Rheological Models ◽  
Implicit Euler Scheme ◽  
Linear Spring ◽  
A Chain ◽  
Uniqueness Theory

A chain sliding on a fixed support, made out of some elementary rheological models (dry friction element and linear spring) can be covered by the existence and uniqueness theory for maximal monotone operators. Several behavior from quasistatic to dynamical are investigated. Moreover, classical results of numerical analysis allow to use a numerical implicit Euler scheme.

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

2021 ◽  
pp. 2250106
Author(s):  
A. A. Mebawondu ◽  
L. O. Jolaoso ◽  
H. A. Abass ◽  
O. K. Narain
Keyword(s):  
Hilbert Spaces ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Split Feasibility Problem ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Inertial Method ◽  
Split Feasibility

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

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