scholarly journals On a second-order functional evolution problem with time and state dependent maximal monotone operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Soumia Saïdi
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Second Order ◽  
Absolutely Continuous ◽  
Lipschitz Continuous ◽  
State Dependent ◽  
Real Separable Hilbert Space ◽  
Order State

<p style='text-indent:20px;'>The present paper proposes, in a real separable Hilbert space, to analyze the existence of solutions for a class of perturbed second-order state-dependent maximal monotone operators with a finite delay. The dependence of the operators is -in some sense- absolutely continuous (or bounded continuous) variation in time, and Lipschitz continuous in state. The approach to solve our problem is based on a discretization scheme. The uniqueness result is applied to optimal control.</p>

ON THE EQUATION OF BARENBLATT–SOBOLEV

2011 ◽  
Vol 13 (05) ◽  
pp. 843-862 ◽  
Author(s):  
ADIMURTHI ◽  
NGONN SEAM ◽  
GUY VALLET
Keyword(s):  
Existence And Uniqueness ◽  
Stochastic Perturbation ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Critical Value ◽  
Sobolev Equation ◽  
Lipschitz Continuous ◽  
Sobolev Problem ◽  
Uniqueness Of The Solution

In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂ut) - Δu - ϵΔ∂ut = g with u(0, ⋅) = u0 where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and [Formula: see text]. We show the existence of a critical value ϵ0 >0 such that: if ϵ > ϵ0, then the problem admits a unique solution; if ϵ = ϵ0, the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ0, then the solution is not unique in general. Passing to the limit with ϵ to 0+, we prove the existence (and uniqueness) of the solution of the Barenblatt differential inclusion Δu + g ∈ f(∂ut) for a class of maximal monotone operators f. Next, we give an extension of the main result for a stochastic perturbation of the problem and we give some numerical illustrations of the Barenblatt and the Barenblatt–Sobolev equation.

scholarly journals Second Order Cones for Maximal Monotone Operators via Representative Functions

2008 ◽  
Vol 16 (2-3) ◽  
pp. 157-184 ◽  
Author(s):  
A. C. Eberhard ◽  
J. M. Borwein
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Second Order

scholarly journals Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

2019 ◽  
Vol 52 (1) ◽  
pp. 274-282
Author(s):  
Behzad Djafari Rouhani ◽  
Mohsen Rahimi Piranfar
Keyword(s):  
Asymptotic Behavior ◽  
Strong Convergence ◽  
Nonlinear Oscillator ◽  
Evolution Equations ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Second Order ◽  
Proximal Point ◽  
Optim Theory

AbstractWe consider the following second order evolution equation modelling a nonlinear oscillator with damping$$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f(t) we show that A−1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A−1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A−1(0). As a discrete version of (SEE), we consider the following second order difference equation$${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].

AN EXISTENCE AND UNIQUENESS RESULT FOR AN ELASTO-PIEZOELECTRIC PROBLEM WITH DAMAGE

2009 ◽  
Vol 19 (01) ◽  
pp. 31-50 ◽  
Author(s):  
JOSÉ R. FERNÁNDEZ ◽  
KENNETH L. KUTTLER
Keyword(s):  
Existence And Uniqueness ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Coupled System ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Comparison Result ◽  
Evolutionary Equations ◽  
Micro Cracks ◽  
Piezoelectric Effects

The aim of this paper is to study the damage evolution in an elasto-piezoelectric body. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, and the piezoelectric effects are included into the model. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is then proved by using the theory of maximal monotone operators, the Schauder fixed-point theorem, and a comparison result.

Evolution problems involving state-dependent maximal monotone operators

2020 ◽  
pp. 1-17 ◽  
Author(s):  
Fatiha Selamnia ◽  
Dalila Azzam-Laouir ◽  
Manuel D. P. Monteiro Marques
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Evolution Problems ◽  
State Dependent

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 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 On the generalized parallel sum of two maximal monotone operators of Gossez type (D)

2012 ◽  
Vol 391 (1) ◽  
pp. 82-98 ◽  
Author(s):  
Radu Ioan Boţ ◽  
Szilárd László
Keyword(s):  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Parallel Sum ◽  
Type D

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.

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