Evolution operators in empathy with a semi-group.
We find an exact asymptotic formula for the singular values of the integral operator of the form (, a Jordan measurable set) where , , and is slowly varying function with some additional properties. The formula is an explicit expression in terms of and .
The liner parabolic equation ∂y ∂t − 1 2 Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝd with smooth boundary is exactly null controllable on each finite interval if 𝒪0is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of x1d4aa; . Here,F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.
The rate of growth of an operator T satisfying the mean ergodic theorem (MET) cannot be faster than linear. It was recently shown (Kornfeld-Kosek, Colloq. Math. 98 (2003)) that for every γ > 0, there are positive L¹[0,1] operators T satisfying MET with . In the class of positive L¹ operators this is the most one can hope for in the sense that for every such operator T, there exists a γ₀ > 0 such that In this note we construct an example of a nonpositive L¹ operator with the highest possible...
A bifurcation problem for variational inequalities is studied, where is a closed convex cone in , , is a matrix, is a small perturbation, a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.