The monodromy maps for the quaternionic Riccati equations with periodic coefficients $\dot{z}=zp(t)z+q(t)z+zr(t)+s(t)$ in $\mathbb H\mathbb P^{1}$ are quternionic Möbius transformations. We prove that, like in the case of automorphisms of $\mathbb C\mathbb P^{1}$, the quaternionic homografies are divided into three classes: hyperbolic, elliptic and parabolic.

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrödinger equation.

We prove the existence of global regular solutions to the Navier-Stokes equations in an axially symmetric domain in $\mathbb{R}^3$ and with boundary slip conditions. We assume that initial angular component of velocity and angular component of the external force and angular derivatives of the cylindrical components of initial velocity and of the external force are sufficiently small in corresponding norms. Then there exists a solution such that velocity belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to $L_{5/2}(\Omega^T)$, and we do not have restrictions on $T$.

By means of Mawhin's continuation theorem, we present some sufficient conditions which guarantee the existence of at least one $T$-periodic solution for a first-order neutral equation with variable parameter. The interest is that the coefficient $c$ is not a constant, which is different from the corresponding ones of past work.

For any compact Lie group $G$, we give a decomposition of the group $\{X,Y\}_G^k$ of (unpointed) stable $G$-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the $G$-homotopy classes in terms of the generalized fixed-point transfer and making use of conormal maps.

Let $f\colon M \to M$ be a fiber-preserving map where $S\to M \to B$ is a bundle and $S$ is a closed surface. We study the abelianized obstruction, which is a cohomology class in dimension 2, to deform $f$ to a fixed point free map by a fiber-preserving homotopy. The vanishing of this obstruction is only a necessary condition in order to have such deformation, but in some cases it is sufficient. We describe this obstruction and we prove that the vanishing of this class is equivalent to the existence of solution of a system of equations over a certain group ring with coefficients given by Fox derivatives.

In infinite dimensional Banach spaces the unit sphere is a lipschitzian retract of the unit ball. We use the space of continuous functions vanishing at a point to provide an example of such retraction having relatively small Lipschitz constant.

The aim of this paper is to present fixed point theorems for multivalued operators $ T\colon X \to P(X)$, on a nonempty set $X$ with two metrics $d$ and $\varrho$, satisfying the following generalized $\varphi$-contraction condition: $$ H_{\varrho}(T(x),T(y))\leq \varphi(M^T(x,y)),\quad \text{for every } x,y \in X, $$ where $$ M^T(x,y):=\max \{ \varrho(x,y),D_{\varrho}(x,T(x)),D_{\varrho}(y,T(y)), 2^{-1} [ D_{\varrho}(x,T(y))+D_{\varrho}(y,T(x)) ]\}. $$

In this paper, an admissibility for topological degree of Herz spaces, Herz-type Besov and Triebel-Lizorkin spaces is given.

Our main interest in this work is to characterize certain operator spaces acting on some important vector-valued function spaces such as $(V_{a})_{c_{0}}^{a\in{\mathbb A}}$, by introducing a new kind basis notion for general Topological vector spaces. Where ${\mathbb A}$ is an infinite set, each $V_{a}$ is a Banach space and $(V_{a})_{c_{0}}^{a\in{\mathbb A}}$ is the linear space of all functions $x\colon{\mathbb A}\rightarrow\bigcup V_{a}$ such that, for each $\varepsilon> 0$, the set $\{a\in{\mathbb A}:\| x_{a}\| > \varepsilon\}$ is finite or empty. This is especially important for the vector-valued sequence spaces $(V_{i})_{c_{0}}^{i\in{\mathbb{N}}}$ because of its fundamental place in the theory of the operator spaces (see, for example, [H. P. Rosenthal, The complete separable extension property, J. Oper. Theory, 43 (2000), no. 2, 329-374]).

We investigate topological Hopf-equivalences in the Morse substitution system. In particular, we give an example of not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system.

In this paper, we rigorously prove the existence of a non-trivial periodic orbit for the nonlinear DDE: $x'(t) = - K \sin(x(t-1))$ for $K=1.6$. We show that the equations for the Fourier coefficients have a solution by computing the local Brouwer degree. This degree can be computed by using a homotopy, and its validity can be proved by checking a finite number of inequalities. Checking these inequalities is done by a computer program.

After the publication of the article, we have noticed that in Lemma 5.11 the hypothesis of injectivity for the map $\psi$ is missing. Hence the lemma has to be corrected by adding the assumption ``$\psi$ is injective on ${\mathcal D}$.'' With this modification we also obtain that $\psi(\Lambda) = \Lambda$ and the proof can be simplified. Without the injectivity assumption, it holds that $\psi|_{\Lambda}$ is semicongiugate to the one-sided $m$-shift (and not to the two-sided as stated). In our applications to ODEs, the function $\psi$ is the Poincaré map (a homeomorphism) and therefore this change doesn't affect the meaningfulness for differential systems. We point out that Lemma 5.11 is considered only with the aim of giving an interpretation about the chaotic dynamics found with our approach. For all the other results, we don't need any injectivity assumption on the maps which are involved.