We establish some abstract convergence and compactness results for families of singularly perturbed semilinear parabolic equations and apply them to reaction-diffusion equations with nonlinear boundary conditions and large diffusion. This refines some previous results of [R. Willie, A semilinear reaction-diffusion system of equations and large diffusion, J. Dynam. Differential Equations 16 (2004), 35-63].

We prove the existence of periodic solutions for a planar non-autonomous Hamiltonian system which is a small perturbation of an autonomous system, in the presence of a non-isochronous period annulus. To this aim we use the Poincaré-Birkhoff fixed point theorem, even if the boundaries of the annulus are neither assumed to be invariant for the Poincaré map, nor to be star-shaped. As a consequence, we show how to deal with the problem of bifurcation of subharmonic solutions near a given nondegenerate periodic solution. In this framework, we only need little regularity assumptions, and we do not need to introduce any Melnikov type functions.

In this paper, the fourth-order singular boundary value problem (BVP) \begin{align*} &u^{(4)}(t)=h(t)f(u(t)),\ \ t\in(0,1),\\ &u(0)=u(1)=u'(0)=u'(1)=0 \end{align*} is considered under some conditions concerning the first characteristic value corresponding to the relevant linear operator, where $h$ is allowed to be singular at both $t=0$ and $t=1$. In particular, $f\colon (-\infty,\infty)\rightarrow (-\infty,\infty)$ may be a sign-changing and unbounded function from below, and it is not also necessary to exist a control function for $f$ from below. The existence results of nontrivial solutions and positive-negative solutions are given by the topological degree theory and the fixed point index theory, respectively.

We give a few sufficient conditions for the existence of periodic solutions of the equation $\dot{z}=\sum_{j=0}^n a_j(t)z^j$. We prove the existence of one up to $n$ periodic solutions and heteroclinic ones.

Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main results in this area are based on a false result and that although we can repair one of these two theorems, there remains doubt on the correctness of the other one. Moreover, we will also show that the notion of an infra-nilmanifold endomorphism itself has not always been interpreted in the same way.

Finally, we define a slightly more general concept of the notion of an infra-nilmanifold endomorphism and explain why this is really the right concept to work with.

We consider the class of weak $(p-2)$-faceless $p$-pseudomanifolds with bounded boundaries and coboundaries. We show that in this class the Betti numbers with $\mathbb{Z}_2$ coefficients may be computed in time $O(n)$ and the $\mathbb{Z}_2$ homology generators in time $O(nm)$ where $n$ denotes the cardinality of the $p$-pseudomanifold on input and $m$ is the number of homology generators.

For every nonempty compact convex subset $K$ of a normed linear space a (unique) point $c_K \in K$, called the generalized Chebyshev center, is distinguished. It is shown that $c_K$ is a common fixed point for the isometry group of the metric space $K$. With use of the generalized Chebyshev centers, the central measure $\mu_X$ of an arbitrary compact metric space $X$ is defined. For a large class of compact metric spaces, including the interval $[0,1]$ and all compact metric groups, another `central' measure is distinguished, which turns out to coincide with the Lebesgue measure and the Haar one for the interval and a compact metric group, respectively. An idea of distinguishing infinitely many points forming a dense subset of an arbitrary compact metric space is also presented.

Let $(X,\Vert \cdot\Vert ) $ be an infinite dimensional real Banach space having a Fréchet differentiable norm and $\varphi\colon \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a gauge function. Denote by $J_{\varphi}\colon X\rightarrow X^{\ast}$ the duality mapping on $X$ corresponding to $\varphi$.

Then, for the Kuratowski measure of noncompactness of $J_{\varphi}$, the following estimate holds: $$ \alpha( J_{\varphi}) \geq \sup\bigg\{ \frac{\varphi(r) }{r}\ \bigg|\ r> 0\bigg\} . $$ In particular, for $-\Delta_{p}\colon W_{0}^{1,p}( \Omega)\rightarrow W^{-1,p'}( \Omega) $, $1< p< \infty$, $l/p+l/p' = 1$, viewed as duality mapping on $W_{0}^{1,p}(\Omega)$, corresponding to the gauge function $\varphi(t)=t^{p-1}$, one has $$ \alpha( -\Delta_{p}) =\begin{cases} 1 & \text{for }p=2,\\ \infty & \text{for }p\in( 1,2) \cup( 2,\infty). \end{cases} $$

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously, when we consider orthonormal $k$-frames in $\mathbb{R}^n$ instead of orthonormal bases (full frames).

We also present new results on simultaneous partition of several measures into parts by $k$ mutually orthogonal hyperplanes.

In the case $k=2$ we relate the Rattray and Makeev type results with the well known embedding problem for projective spaces.

It is known that the unit ball in infinitely dimensional Hilbert space can be retracted onto its boundary via a lipschitzian mapping. The magnitude of Lipschitz constant is only roughly estimated. The note contains a number of observations connected to this result and opens some new problems.