Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ defined by the rule: $(f,h)\mapsto f\circ h$ for $f\in \mathcal{C}^{\infty}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$. Denote by $\mathcal{F}(M)$ the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisting of functions $f\colon M\to\mathbb{R}$ taking constant values on connected components of $\partial{M}$, having no critical points on $\partial{M}$, and such that at each of its critical points $z$ the function $f$ is $\mathcal{C}^{\infty}$ equivalent to some homogenenous polynomial without multiple factors. In particular, $\mathcal{F}(M)$ contains all Morse maps. Let also $\mathcal{O}(f) = \{ f\circ h \mid h\in\mathcal{D}(M) \}$ be the orbit of $f$. Previously, the algebraic structure of $\pi_1\mathcal{O}(f)$ was computed for all $f\in\mathcal{F}(M)$, where $M$ is any orientable compact surface distinct from $2$-sphere. In the present paper we compute the group $\pi_0\mathcal{S}(f,\partial\mathbb{M})$, where $\mathbb{M}$ is a Möbius band, and $\mathcal{S}(f,\partial\mathbb{M}) = \{ h\in\mathcal{D}(\mathbb{M}) \mid f\circ h = f, \ h|_{\partial \mathbb{M}} = \mathrm{id}_{\mathbb{M}}\}$ is the subgroup of the corresponding stabilizer of $f$ consisting of diffeomorphisms fixed on the boundary $\partial \mathbb{M}$. As a consequence we obtain an explicit algebraic description of $\pi_1\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein bottle and projective plane.

In this paper, we consider the family of non-autonomous dynamical systems obtained by iterating tent maps with a flat top of constant value $u$ introduced at instants $i$ such that $s_i=0$, where $s$ is a binary sequence that we call the iteration pattern. We introduce symbolic dynamics and study the kneading invariants for these dynamical systems. More precisely, we define the kneading invariants $K(u,s)$ as symbolic sequences and study sufficient conditions for a symbolic sequence to be a kneading invariant $K(u,s)$ for some $u$, for each iteration pattern $s$. Finally, we describe the parameters $u$ for which there are Milnor attractors for iteration patterns $s$ such that $s_{np}=0$ for all $n$, as limits of parameter sequences corresponding to local attractors, organized according to a period-incrementing structure.

In this work, we introduce and explore a rescaled theory of local stable and unstable sets for rescaled-expansive flows and its applications to topological entropy. We introduce a rescaled version of the local unstable sets and the unstable points. We find conditions for points of the phase space to exhibit non-trivial connected pieces of such unstable sets. We apply these results to the problem of proving positive topological entropy for rescaled-expansive flows with non-singular Lyapunov stable sets.

In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form\begin{equation*} x_{k+1}=A_kx_k+f(k,x_k), \quad k\in\mathbb N_0:=\{0,1,\dots\}, \end{equation*} subject to boundary conditions \begin{equation*} \sum_{k=0}^\infty B_kx_k=G(x). \end{equation*} We will show that under appropriate assumptions on each $A_k$, $f$, each $B_k$, and $G$, $\ell^p\big(\mathbb N_0,\mathbb R^{n}\big)$ solutions will exist for $p\in [1,\infty]$.

We prove Brézis-Kato regularity type results for solutions of the higher order nonlinear elliptic equation\[ L u = g(x,u)\quad\text{in }\Omega\]with an elliptic operator $L$ of $2m$ order with variable coefficients and a Carathéodory function $g\colon \Omega\times \mathbb C\to\mathbb C$, where $\Omega\subset\mathbb R^N$ is an open set with $N > 2m$.

In this paper, we prove the existence of at least $N$ pairs of nontrivial solutions to the doubly critical quasilinear elliptic problem \[ -\Delta_p u - \frac{\lambda}{|x|^p}|u|^{p-2}u=a(x)|u|^{p-2}u + |u|^{p^*-2}u \] in $\mathbb{R}^N$, as well as in smooth bounded domains, where $1< p< N$, $0< \lambda< (({N-p})/{p})^p$ and $a$ is strictly positive in a small ball. Our results hold under the assumption that $N\ge p^2$ and $\lambda$ and $\|a^+\|_{L^{N/p}}$ are small enough. To circumvent difficulties due to the lack of compactness of the problem, we combine Krasnosel'skiĭ's genus with a recent classification result by Oliva, Sciunzi, Vaira (J. Math. Pures Appl. 140 (2020), 89-109) and global compactness results by Li, Guo, Niu (Nonlinear Anal. 74 (2011), no. 4, 1445-1464).

In this paper, we study the existence and multiplicity of periodic solutions for singular Duffing equations $x''+g(x)=p(t)$. When $g$ satisfies one-sided Lipschitz condition and the related time map satisfies oscillation property, we prove that the given equation has infinitely many periodic solutions.

Let $f$ be an orientation-preserving homeomorphism of the 2-disc $\mathbb{D}^2$ that fixes the boundary pointwise and leaves invariant a finite subset in the interior of $\mathbb{D}^2$. We study the strong Nielsen equivalence of periodic points of such homeomorphisms $f$ and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang-Zheng, deducing that the obtained forced periodic orbits belong to different strong Nielsen classes.

This paper presents a study, based on Conley's theory, of continuous flows on singular surfaces that admit singularities such as $n$-sheet cones, $n$-sheet cross-caps, $n$-sheet double crossings, $n$-sheet triple crossings, and mixed Gutierrez-Sotomayor (GS) singularities. These flows are referred to as generalized Gutierrez-Sotomayor (GS) flows. The Conley index for each type of singularity is computed. Furthermore, the results from previous works on the local and global existence of Lyapunov functions are extended to encompass generalized GS singularities. Necessary and sufficient conditions for defining a generalized GS flow on an isolating block are established. Additionally, an alternative formula, expressed in terms of a deeper dynamical perspective, for computing the Euler-Poincaré characteristic of generalized GS manifolds is introduced.

Applying techniques originally developed for systems lacking a variational structure, we establish conditions for the existence of solutions in systems that possess this property but their energy functional is unbounded both above and below. We show that, in general, our conditions differ from those in the classical mountain pass approach by Ambrosetti-Rabinowitz when dealing with systems of this type. Our theory is put into practice in the context of a coupled system of Stokes equations with reaction terms, where we establish sufficient conditions for the existence of a solution. The systems under study are intermediary between gradient-type systems and Hamiltonian systems.