In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) $u$ for which $Ex(u,n)=O\left(n\right)$. The function $Ex(u,n)$ measures the maximum length of finite sequences over $n$ symbols which contain no subsequence of the type $u$. It follows from the result of Hart and Sharir that the containment $ababa\prec u$ is a (minimal) obstacle to $Ex(u,n)=O\left(n\right)$. We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate...

For a space $Z$, we denote by $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\le 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $ℱ\left(Z\right)$, $𝒦\left(Z\right)$ and ${ℱ}_2\left(Z\right)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable...

Values of $\lambda$ are determined for which there exist positive solutions of the system of three-point boundary value problems, $u^{\text{'}\text{'}}+\lambda a\left(t\right)f\left(v\right)=0$, $v^{\text{'}\text{'}}+\lambda b\left(t\right)g\left(u\right)=0$, for $0<t<1$, and satisfying, $u\left(0\right)=\beta u\left(\eta \right)$, $u\left(1\right)=\alpha u\left(\eta \right)$, $v\left(0\right)=\beta v\left(\eta \right)$, $v\left(1\right)=\alpha v\left(\eta \right)$. A Guo-Krasnosel’skii fixed point theorem is applied.

We shall prove the following statements: Given a sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ in a Banach space $𝐗$ enjoying the weak Banach-Saks property, there is a subsequence (or a permutation) ${\left\{b_n\right\}}_{n=1}^{\infty }$ of the sequence ${\left\{a_n\right\}}_{n=1}^{\infty }$ such that $$\underset{n\to \infty }{lim}\frac{1}{n}\sum _{j=1}^n{b}_j=a$$ whenever $a$ belongs to the closed convex hull of the set of weak limit points of ${\left\{a_n\right\}}_{n=1}^{\infty }$. In case $𝐗$ has the Banach-Saks property and ${\left\{a_n\right\}}_{n=1}^{\infty }$ is bounded the converse assertion holds too. A characterization of reflexive spaces in terms of limit points and cores of bounded sequences is also given. The motivation...