A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

In this note we prove the existence of radially symmetric solutions for a class of fractional Schrödinger equations in $\mathbb{R}^N$ of the form \begin{equation*} (-\Delta)^s u + V(x) u = g(u), \end{equation*} where the nonlinearity $g$ does not satisfy the usual Ambrosetti-Rabinowitz condition. Our approach is variational in nature, and leans on a Pohozaev identity for the fractional laplacian.

For a topological action $\Phi$ of a countable amenable orderable group $G$ on a compact metric space we introduce a concept of the asymptotic relation $\mathbf{A} (\Phi)$ and we show that $\mathbf{A} (\Phi)$ is non-trivial if the topological entropy $h(\Phi)$ is positive. It is also proved that if the Pinsker $\sigma$-algebra $\pi_{\mu}(\Phi)$ is trivial, where $\mu$ is an invariant measure with full support, then $\mathbf{A} (\Phi)$ is dense. These results are generalizations of those of Blanchard, Host and Ruette ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) that concern the asymptotic relation for $\mathbb{Z}$-actions.

We give an example of an expansive $G$-action ($G=\mathbb{Z}^2$) with $\mathbf{A} (\Phi)$ trivial which shows that the Bryant-Walters classical result ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) fails to be true in general case.

We study second-order ordinary differential equations of Newtonian type. The forcing terms under consideration are the product of a nonlinearity which is singular at the origin with an indefinite weight. Under some additional assumptions we show the existence of periodic solutions.

In this paper we establish existence of connected components of positive solutions of the equation $ -\Delta_{p} u = \lambda f(u)$ in $\Omega$, under Dirichlet boundary conditions, where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\Delta_{p}$ is the $p$-Laplacian, and $f \colon (0,\infty) \rightarrow \mathbb{R} $ is a continuous function which may blow up to $\pm \infty$ at the origin.

We give an algebraic proof of the Theorem of Cheng Ye You that the least number of $n$-periodic points, in the continuous homotopy class of a self-map of a torus, can be realized by a smooth map.

In this work we give some equivalent formulations for the optimization problem \begin{multline*} \max\bigg\{ \int_{\Omega} \xi \,d\mu + \int_{\Gamma_{N}}\xi \,d\nu;\ \xi \in W^{1,\infty}(\Omega) \text{ such that } \\ \xi_{/\Gamma_{D}}= 0,\ |\nabla\xi(x)|\leq 1 \text{ a.e. } x\in \Omega\bigg\}, \end{multline*} where the boundary of $\Omega$ is $\Gamma=\Gamma_{N}\cup\Gamma_{D}$.

In this paper, a competitor-competitor-mutualist model with cross-diffusion is studied by means of the Leray-Schauder degree theory and global bifurcation theory. The conditions for the existence and multiplicity of positive solutions are established.

We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits.

We prove a bifurcation and multiplicity result that is independent of the dimension $N$ for a critical $p$-Laplacian problem that is an analog of the Brezis-Nirenberg problem for the quasilinear case. This extends a result in the literature for the semilinear case $p = 2$ to all $p \in (1,\infty)$. In particular, it gives a new existence result when $N \lt p^2$. When $p \ne 2$ the nonlinear operator $- \Delta_p$ has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo-index related to the $\mathbb Z_2$-cohomological index that is applicable here.

In this paper, by introducing a new concept of the stable $(g,f,\alpha)$-quasimonotonicity and applying the properties of Clarke's generalized gradient and KKM technique, we show the existence results of solutions for hemivariational inequalities when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. In the last section, we also give an example to present the our main result.

In this paper, Morse theory is used to establish the existence of multiple solutions for an impulsive boundary value problem posed on the half-line.

In this paper, we are concerned with existence and asymptotic behavior of ground state in the whole space $\mathbb{R}^3$ for quasilinear Schrödinger-Poisson systems $$ \begin{cases} -\Delta u+u+K(x)\phi(x)u=a(x)f(u), & x\in \mathbb{R}^3, \\ -\mbox{div}[(1+\varepsilon^4|\nabla\phi|^2)\nabla\phi]=K(x)u^2, & x\in \mathbb{R}^3, \end{cases} $$ when the nonlinearity coefficient $\varepsilon\gt 0$ goes to zero, where $f(t)$ is asymptotically linear with respect to $t$ at infinity. Under appropriate assumptions on $K$, $a$ and $ f$, we establish existence of a ground state solution $(u_\varepsilon, \phi_{\varepsilon, K}(u_\varepsilon))$ of the above system. Furthermore, for all $\varepsilon$ sufficiently small, we show that $(u_\varepsilon, \phi_{\varepsilon, K}(u_\varepsilon))$ converges to $(u_0, \phi_{0, K}(u_0))$ which is the solution of the corresponding system for $\varepsilon=0$.

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.

We consider a nonlinear Neumann problem driven by the $p$-Laplacian with a concave parametric reaction term and an asymptotically linear perturbation. We prove a multiplicity theorem producing five nontrivial solutions all with sign information when the parameter is small. For the semilinear case $(p=2)$ we produce six solutions, but we are unable to determine the sign of the sixth solution. Our approach uses critical point theory, truncation and comparison techniques, and Morse theory.

In this paper we study the solvability of a nonlinear Hammerstein type integral equation in the space of functions of bounded Riesz-Medvedev variation. To this end, we derive a compactness criterion and apply Schauder's fixed point theorem to a suitable operator whose fixed points coincide with the solutions of the integral equation.

In this paper we introduce a concept of possibly infinite self-similar system which generalizes the attractor of a possibly infinite iterated function system whose constitutive functions are $\varphi$-contractions. We prove that for a uniformly possibly infinite self-similar system there exists a remetrization which makes contractive all its constitutive functions. Then, based on this result, we show that for such a system there exist a comparison function $\varphi $ and a remetrization of the system which makes $\varphi $-contractive all its constitutive functions. Finally we point out that in the case of a finite set of constitutive functions our concept of a possibly infinite self-similar system coincides with Kameyama's concept of a topological self-similar system.

Every expanding map on a closed manifold is topologically conjugate to an expanding map on an infra-nilmanifold, but not every infra-nilmanifold admits an expanding map. In this article we give a complete algebraic characterization of the infra-nilmanifolds admitting an expanding map. We show that, just as in the case of Anosov diffeomorphisms, the existence of an expanding map depends only on the rational holonomy representation of the infra-nilmanifold. A similar characterization is also given for the infra-nilmanifolds with a non-trivial self-cover, which corresponds to determining which almost-Bieberbach groups are co-Hopfian. These results provide many new examples of infra-nilmanifolds without non-trivial self-covers or expanding maps.

A semilinear version of parabolic-elliptic Keller-Segel system with the critical nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large initial datum, provided $r \gt 1$. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to $\sim {1}/({1-r})$.

In the paper we give a generalization of the Filippov-Ważewski Theorem to the second order differential inclusions \begin{equation} \mathcal{D}y=y^{\prime \prime }-A^{2}y\in F( t,y) , \tag{$*$} \end{equation} with the initial conditions \begin{equation} y( 0) =\alpha ,\quad y^{\prime }( 0) =\beta , \tag{$**$} \end{equation} where $A\in \mathbb{R}^{d\times d}$ and $F\colon [ 0,T] \times \mathbb{R}^{d}\leadsto c( \mathbb{R}^{d}) $ is a multifunction satisfying for each $t\in [ 0,T] $ the Lipschitz condition in $y$ \begin{equation*} d_{H}( F( t,y_{1}) ,F( t,y_{2}) ) \leq l( t) \vert y_{1}-y_{2}\vert , \end{equation*} where $l(\,\cdot\,) $ is integrable. The main result is the following:

Theorem 5.1. Assume that $F\colon[ 0,T] \times \mathbb{R} ^{d}\leadsto c( \mathbb{R}^{d}) $ is measurable in $t$, Lipschitz continuous in $x\in \mathbb{R}^{d}$ $($with integrable constant$)$ and integrably bounded. Let $r\in W^{2,1}$ be a solution of the relaxed problem \begin{equation} \mathcal{D}y=y^{\prime \prime }-A^{2}y\in {\rm cl}\, {\rm co}\,F ( t,y) , \tag{$**$$*$} \end{equation} with $(**)$. Then, for each $\varepsilon \gt 0$, there exists a solution $y\in W^{2,1}$ of $(*)$ with $(**)$ such that \begin{equation*} \Vert y-r\Vert _{C^{1}[ 0,T] } \lt \varepsilon . \end{equation*}

The proof goes via a version of the Fillipov Lemma (Theorem 4.4) for inclusions ($*$).