Let X be a real separable Banach space. The boundary value problem x′ ∈ A(t)x + F (t, x), t ∈ R+, Ux = a, (B) is studied on the infinite interval R+ = [0,∞). Here, the closed and densely defined linear operator A(t) : X ⊃ D(A)→ X, t ∈ R+, generates an evolution operator W (t, s). The function F : R+×X → 2X is measurable in its first variable, upper semicontinuous in its second and has weakly co...

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized ...

Abstract We study mean equicontinuous actions of locally compact σ-compact amenable groups on metric spaces. In this setting, we establish the equivalence equicontinuity and topo-isomorphy to maximal factor provide a characterization an action via properties its product. This enables us show weaker notion Besicovitch-mean in fairly high generality, including abelian as well minimal general grou...

For an arbitrary topological group G any compact G-dynamical system (G, X) can be linearly G-represented as a weak∗-compact subset of a dual Banach space V ∗. As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP). In this paper we study the wider class of compact G-systems which can be linearly represented as a weak∗-c...

Let $(X,T)$ be a topological dynamical system and $n\geq 2$. We say that is $n$-tuplewise IP-sensitive (resp. thickly sensitive) if there exists constant $\delta>0$ with the property for each non-empty open subset $U$ of $X$, exist $x_1,x_2,\dotsc,x_n\in U$ such \[ \Bigl\{k\in\mathbb{N}\colon \min_{1\le i\delta\Bigr\} \] an IP-set thick set). obtain several sufficient...

Firstly, it is introduced that the concepts of G-almost periodic point and G-sequence shadowing property. Then, we discuss dynamical relationship between sequence map {gk}∞k=1 limit g under G-strongly uniform convergence topological group action. We can get (1) Let be converge to where equicontinuous {yk}∞k=1 {gk}∞k=1. If limk → ∞ yk = y, then y an g; (2) are equicontinuous, limsup APG(gk) ⊂ AP...

We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.