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Equilibria and strict equilibria of multivalued maps on noninvariant sets

Pierre Cardaliaguet, Grzegorz Gabor, Marc Quincampoix (2003)

Annales Polonici Mathematici

This paper is concerned with existence of equilibrium of a set-valued map in a given compact subset of a finite-dimensional space. Previously known conditions ensuring existence of equilibrium imply that the set is either invariant or viable for the differential inclusion generated by the set-valued map. We obtain some equilibrium existence results with conditions which imply neither invariance nor viability of the given set. The problem of existence of strict equilibria is also discussed.

Equivalent formulation and numerical analysis of a fire confinement problem

Alberto Bressan, Tao Wang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of variational problems for differential inclusions, related to the control of wild fires. The area burned by the fire at time t> 0 is modelled as the reachable set for a differential inclusion $\dot{x}$ ∈F(x), starting from an initial set R0. To block the fire, a barrier can be constructed progressively in time. For each t> 0, the portion of the wall constructed within time t is described by a rectifiable set γ(t) ⊂ $ℝ^{2}$ . In this paper we show that the search for blocking strategies...

Equivariant degree of convex-valued maps applied to set-valued BVP

Zdzisław Dzedzej (2012)

Open Mathematics

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.

Errata to the paper: ``Regularization of closed-valued multifunctions in a non-metric setting''

Diego Averna (1995)

Mathematica Slovaca

Euler approximation of nonconvex discontinuous differential inclusions.

Donchev, Tzanko (2002)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

Euler-Lagrange inclusions and existence of minimizers for a class of non-coercive variational problems.

Crasta, Graziano, Malusa, Annalisa (2000)

Journal of Convex Analysis

Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms

Thomas Lorenz (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space...

Evolution inclusions in non separable Banach spaces

Francesco Saverio De Blasi, Giulio Pianigiani (1999)

Commentationes Mathematicae Universitatis Carolinae

We study a Cauchy problem for non-convex valued evolution inclusions in non separable Banach spaces under Filippov type assumptions. We establish existence and relaxation theorems.

Evolution inclusions of the subdifferential type depending on a parameter.

Kandilakis, Dimitrios, Papageorgiou, Nikolaos S. (1992)

Commentationes Mathematicae Universitatis Carolinae

Evolution inclusions of the subdifferential type depending on a parameter

Dimitrios A. Kandilakis, Nikolaos S. Papageorgiou (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study evolution inclusions generated by time dependent convex subdifferentials, with the orientor field $F$ depending on a parameter. Under reasonable hypotheses on the data, we show that the solution set $S (λ)$ is both Vietoris and Hausdorff metric continuous in $λ \in Λ$ . Using these results, we study the variational stability of a class of nonlinear parabolic optimal control problems.

Evolution problems associated with nonconvex closed moving sets with bounded variation.

Castaing, Charles, Monteiro Marques, Manuel D.P. (1996)

Portugaliae Mathematica

Existence and approximation results for gradient flows

Riccarda Rossi, Giuseppe Savaré (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$ $\{\begin{matrix} u^{...} \end{matrix}$

Existence and controllability for nondensely defined partial neutral functional differential inclusions

Khalil Ezzinbi, Soumia Lalaoui Rhali (2015) Applications of Mathematics

We give sufficient conditions for the existence of integral solutions for a class of neutral functional differential inclusions. The assumptions on the generator are reduced by considering nondensely defined Hille-Yosida operators. Existence and controllability results are established by combining the theory of addmissible multivalued contractions and Frigon's fixed point theorem. These results are applied to a neutral partial differential inclusion with diffusion.

Existence and density results for retarded subdifferential evolution inclusions

Tiziana Cardinali, Simona Pieri (1996) Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem")....

Existence and relaxation results for nonlinear evolution inclusions revisited.

Migórski, S. (1995) Journal of Applied Mathematics and Stochastic Analysis

Existence and relaxation results for nonlinear second order evolution inclusions

Stanisław Migórski (1995) Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we study nonlinear evolution inclusions associated with second order equations defined on an evolution triple. We prove two existence theorems for the cases where the orientor field is convex valued and nonconvex valued, respectively. We show that when the orientor field is Lipschitzean, then the set of solutions of the nonconvex problem is dense in the set of solutions of the convexified problem.

Existence and uniqueness for dynamical unilateral contact with Coulomb friction : a model problem

Patrick Ballard, Stéphanie Basseville (2005) ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A simple dynamical problem involving unilateral contact and dry friction of Coulomb type is considered as an archetype. We are concerned with the existence and uniqueness of solutions of the system with Cauchy data. In the frictionless case, it is known [Schatzman, Nonlinear Anal. Theory, Methods Appl. 2 (1978) 355–373] that pathologies of non-uniqueness can exist, even if all the data are of class

C^{\infty}

. However, uniqueness is recovered provided that the data are analytic [Ballard, Arch. Rational Mech....

Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem

Patrick Ballard, Stéphanie Basseville (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A simple dynamical problem involving unilateral contact and dry friction of Coulomb type is considered as an archetype. We are concerned with the existence and uniqueness of solutions of the system with Cauchy data. In the frictionless case, it is known [Schatzman, Nonlinear Anal. Theory, Methods Appl.2 (1978) 355–373] that pathologies of non-uniqueness can exist, even if all the data are of class C∞. However, uniqueness is recovered provided that the data are analytic [Ballard, Arch. Rational...

Existence and uniqueness of solutions to dynamical unilateral contact problems with coulomb friction: the case of a collection of points

Alexandre Charles, Patrick Ballard (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This study deals with the existence and uniqueness of solutions to dynamical problems of finite freedom involving unilateral contact and Coulomb friction. In the frictionless case, it has been established [P. Ballard, Arch. Rational Mech. Anal. 154 (2000) 199–274] that the existence and uniqueness of a solution to the Cauchy problem can be proved under the assumption that the data are analytic, but not if they are assumed to be only of class C∞. Some years ago, this finding was extended [P. Ballard...

Existence of global solutions to differential inclusions; a priori bounds

Ovidiu Cârjă, Alina Ilinca Lazu (2012)

Mathematica Bohemica

The paper presents an existence result for global solutions to the finite dimensional differential inclusion $y^{'} \in F (y),$ $F$ being defined on a closed set $K .$ A priori bounds for such solutions are provided.

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