From the KKM theorem for the “closed” and “open” valued cases, we deduce a generalization of the Alexandroff–Pasynkoff theorem, existence theorems for almost fixed points of lower semicontinuous multimaps, and a partial solution of the Ben-El-Mechaiekh conjecture.

We show the lower semicontinuity of the family of pullback attractors for the singularly nonautonomous plate equation with structural damping utt + a(t, x)ut + (−∆)ut + (−∆)u+ λu = f(u), in the energy space H2 0 (Ω)×L2(Ω) under small perturbations of the damping term a.

In most problems, the ideal situation is having the asymptotic dynamics of one equation the same as the asymptotic dynamics of its discretization. Although, when we are studying the linear wave equation, we note that the spectrum of discretization and the spectrum of its continuous counterpart are far away from each other, no matter how fine the discretization is. This fact is restrictive in th...

If K is a nonempty compact subset of X and f : X → [−∞,∞] is lower semicontinuous, there is some x0 ∈ K such that f(x0) = infx∈K f(x). A rate function is a function I : X → [0,∞] such that (i) there is some x ∈ X with I(x) <∞ and (ii) for any c ∈ [0,∞], f−1([0, c]) is compact (namely, the sublevel sets of I are compact). Condition (ii) implies that a rate function is lower semicontinuous. For S...

RECENTLY G. C. Young* and A. Denjoyf have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus."$ However, instead of confining our...

For a class of operator inclusions with multivalued terms fulllling mixed semicontinuity hypotheses, the existence of solutions is established by chieey using Bressan and Colombo's directionally continuous selection theorem as well as Ky Fan's xed point principle. An application to the Cauchy problem is then performed.

Inspired by the notion of biconic semi-copulas, we introduce biconic semi-copulas with a given section. Such semi-copulas are constructed by linear interpolation on segments connecting the graph of a continuous and decreasing function to the points (0, 0) and (1, 1). Special classes of biconic semi-copulas with a given section such as biconic (quasi-)copulas with a given section are considered....

4. Let (Ω,Σ, μ) be a measure space. If f : Ω → R, then we say that f is a measurable function if for every t ∈ R, the level set: Sf (t) = {x ∈ Ω|f(x) > t} is measurable. f : Ω→ C is measurable if and only if both Re(f) and Im(f) are measurable. 5. A real function f : Ω → R is lower semicontinuous on Ω if Sf (t) is open, and is upper semicontinuous if {x ∈ Ω|f(x) < t} is open. Equivalently, f is...

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.