On the virtual convexity of the domain and range of a nonlinear maximal monotone operator

1970 ◽  
Vol 185 (2) ◽  
pp. 81-90 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone ◽  
Nonlinear Maximal Monotone ◽  
Nonlinear Maximal Monotone Operator

scholarly journals Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1395
Author(s):  
Charles Castaing ◽  
Christiane Godet-Thobie ◽  
Le Xuan Truong
Keyword(s):  
Fractional Order ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Separable Hilbert Space ◽  
Maximal Monotone ◽  
Fractional Flow ◽  
Order Problem ◽  
Absolutely Continuous ◽  
State Dependent ◽  
Fractional Differential

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

scholarly journals NOTES ON GRAPH-CONVERGENCE FOR MAXIMAL MONOTONE OPERATORS

2010 ◽  
Vol 83 (1) ◽  
pp. 22-29 ◽  
Author(s):  
FILOMENA CIANCIARUSO ◽  
GIUSEPPE MARINO ◽  
LUIGI MUGLIA ◽  
HONG-KUN XU
Keyword(s):  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Null Sequence ◽  
Graph Convergence ◽  
Common Domain

AbstractWe construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

scholarly journals The convex function determined by a multifunction

1996 ◽  
Vol 54 (1) ◽  
pp. 87-97 ◽  
Author(s):  
M. Coodey ◽  
S. Simons
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Convex Function ◽  
Monotone Operator ◽  
Open Problem ◽  
Maximal Monotone Operator ◽  
Convex Functions ◽  
Maximal Monotone ◽  
Minimax Theorem ◽  
Local Boundedness

We shall show how each multifunction on a Banach space determines a convex function that gives a considerable amount of information about the structure of the multifunction. Using standard results on convex functions and a standard minimax theorem, we strengthen known results on the local boundedness of a monotone operator, and the convexity of the interior and closure of the domain of a maximal monotone operator. In addition, we prove that any point surrounded by (in a sense made precise) the convex hull of the domain of a maximal monotone operator is automatically in the interior of the domain, thus settling an open problem.

scholarly journals A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-31 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Maximal Monotone ◽  
Generalized Mixed Equilibrium Problem ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

scholarly journals Optimal Control of Plasticity with Inertia

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Stephan Walther
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Monotone Operator ◽  
Maximal Monotone Operator ◽  
Kinematic Hardening ◽  
Maximal Monotone ◽  
Small Strain ◽  
State Equation ◽  
Elasto Plasticity

The paper is concerned with an optimal control problem governed by the equations of elasto plasticity with linear kinematic hardening and the inertia term at small strain. The objective is to optimize the displacement field and plastic strain by controlling volume forces. The idea given in [10] is used to transform the state equation into an evolution variational inequality (EVI) involving a certain maximal monotone operator. Results from [27] are then used to analyze the EVI. A regularization is obtained via the Yosida approximation of the maximal monotone operator, this approximation is smoothed further to derive optimality conditions for the smoothed optimal control problem.

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