Continuity of hysteresis operators in Sobolev spaces

Pavel Krejčí; Vladimír Lovicar

Displaying similar documents to “Continuity of hysteresis operators in Sobolev spaces”

A remark on the local Lipschitz continuity of vector hysteresis operators

Pavel Krejčí (2001)

Applications of Mathematics

Similarity:

It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^{1}$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$ . We prove that in the regular case, this condition is also necessary.

Hysteresis memory preserving operators

Pavel Krejčí (1991)

Applications of Mathematics

Similarity:

The recent development of mathematical methods of investigation of problems with hysteresis has shown that the structure of the hysteresis memory plays a substantial role. In this paper we characterize the hysteresis operators which exhibit a memory effect of the Preisach type (memory preserving operators). We investigate their properties (continuity, invertibility) and we establish some relations between special classes of such operators (Preisach, Ishlinskii and Nemytskii operators)....

On the existence of viable solutions for a class of second order differential inclusions

Aurelian Cernea (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

We prove the existence of viable solutions to the Cauchy problem x” ∈ F(x,x’), x(0) = x₀, x’(0) = y₀, where F is a set-valued map defined on a locally compact set $M \subset R^{2 n}$ , contained in the Fréchet subdifferential of a ϕ-convex function of order two.

Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

Similarity:

Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

Homogenization of diffusion equation with scalar hysteresis operator

Jan Franců (2001)

Mathematica Bohemica

Similarity:

The paper deals with a scalar diffusion equation $c u_{t} = {(F [u_{x}])}_{x} + f,$ where $F$ is a Prandtl-Ishlinskii operator and $c, f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially...

Local Lipschitz continuity of the stop operator

Wolfgang Desch (1998)

Applications of Mathematics

Similarity:

On a closed convex set $Z$ in $ℝ^{N}$ with sufficiently smooth ( $𝒲^{2, \infty}$ ) boundary, the stop operator is locally Lipschitz continuous from $𝐖^{1, 1} ([0, T], ℝ^{N}) \times Z$ into $𝐖^{1, 1} ([0, T], ℝ^{N})$ . The smoothness of the boundary is essential: A counterexample shows that $𝒞^{1}$ -smoothness is not sufficient.

Biseparating maps on generalized Lipschitz spaces

Denny H. Leung (2010)

Studia Mathematica

Similarity:

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are...