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A Tikhonov-type theorem for abstract parabolic differential inclusions in Banach spaces

Anastasie Gudovich, Mikhail Kamenski, Paolo Nistri (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for ε ≥ 0, where ε is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset Z L ( ε ) of the solution set of the singularly perturbed system. This subset is the set of...

Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs

Takeshi Fukao, Nobuyuki Kenmochi (2014)

Mathematica Bohemica

Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize...

An existence result for impulsive functional differential inclusions in Banach spaces

Irene Benedetti (2004)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We use the topological degree theory for condensing multimaps to present an existence result for impulsive semilinear functional differential inclusions in Banach spaces. Moreover, under some additional assumptions we prove the compactness of the solution set.

Anti-periodic solutions to a parabolic hemivariational inequality

Jong Yeoul Park, Hyun Min Kim, Sun Hye Park (2004)

Kybernetika

In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form b ( u ) , where b ∈ L loc ∞ ( R ) . Extending b to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality.

Approximate weak invariance for semilinear differential inclusions in Banach spaces

Alina Lazu, Victor Postolache (2011)

Open Mathematics

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C 0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + c o ¯ F(x(t)), without any Lipschitz conditions...

Boundary value problem for differential inclusions in Fréchet spaces with multiple solutions of the homogeneous problem

Irene Benedetti, Luisa Malaguti, Valentina Taddei (2011)

Mathematica Bohemica

The paper deals with the multivalued boundary value problem x ' ∈ A ( t , x ) x + F ( t , x ) for a.a. t ∈ [ a , b ] , M x ( a ) + N x ( b ) = 0 , in a separable, reflexive Banach space E . The nonlinearity F is weakly upper semicontinuous in x . We prove the existence of global solutions in the Sobolev space W 1 , p ( [ a , b ] , E ) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.

BV solutions of rate independent differential inclusions

Pavel Krejčí, Vincenzo Recupero (2014)

Mathematica Bohemica

We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For BV (bounded variation) data we compare different notions of BV solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case...

Continuity of the non-convex play operator in the space of rectifiable curves

Jana Kopfová, Vincenzo Recupero (2023)

Applications of Mathematics

We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the B V -norm and to the B V -strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent.

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