The main purpose of this work is to study coincidences of fibre-preserving self-maps over the circle $S^1$ for spaces which are fibre bundles over $S^1$ and the fibre is the torus $T$. We classify all pairs of self-maps over $S^1$ which can be deformed fibrewise to a pair of coincidence free maps.

In this paper we consider the existence of periodic solutions of one-parameter and two-parameter families of second order singular differential equations.

In this work we study the continuity for the family of global attractors of the equations $u_{tt}-\Delta u-\Delta u_t-\varepsilon \Delta u_{tt}=f(u)$ at $\varepsilon=0$ when $\Omega$ is a bounded smooth domain of $\mathbb{R}^n$, with $n\geq 3$, and the nonlinearity $f$ satisfies a subcritical growth condition. Also, we obtain an uniform bound for the fractal dimension of these global attractors.

We study reasons related to two-dimensional CW-complexes which prevent an extension of the Hopf–Whitney Classification Theorem for maps from those complexes into the real projective plane, even in the simpler situation in which the complex has trivial second integer cohomology group. We conclude that for such a two-complex $K$, the following assertions are equivalent: (1) Every based map from $K$ into the real projective plane is based homotopic to a constant map; (2) The skeleton pair $(K,K^1)$ is homotopy equivalent to that of a model two-complex induced by a balanced group presentation; (3) The number of two-dimensional cells of $K$ is equal to the first Betti number of its one-skeleton; (4) $K$ is acyclic; (5) Every based map from $K$ into the circle $S^1$ is based homotopic to a constant map.

In this paper by using the minimal principle and Morse theory, we prove the existence of solutions to the following Kirchhoff type fractional differential equation: \begin{equation*} \begin{cases} \displaystyle M \bigg (\int_{\mathbb{R}} (|{}_{- \infty} D_t^\alpha u (t)|^2 + b (t) |u(t)|^2 )\, d t \bigg) \\ \qquad \cdot ({}_tD_\infty^{\alpha} ({}_{- \infty} D_t^\alpha u (t) ) + b(t) u (t)) = f (t, u (t)), & t \in \mathbb{R}, \\ u \in H^\alpha (\mathbb{R}), \end{cases} \end{equation*} where $\alpha \in ({1}/{2},1)$, ${}_tD_\infty^{\alpha}$ and ${}_{- \infty} D_t^\alpha$ are the right and left inverse operators of the corresponding Liouville-Weyl fractional integrals of order $\alpha$ respectively, $H^\alpha$ is the classical fractional Sobolev Space, $u \in \mathbb{R}$, $b \colon \mathbb{R} \to \mathbb{R}$, $\inf\limits_{t \in \mathbb{R}} b (t) \gt 0$, $f \colon \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ Carathéodory function and $M\colon \mathbb{R}^+ \to \mathbb{R}^+$ is a function that satisfy some suitable conditions.

We consider a $T$-periodically perturbed autonomous functional differential equation of neutral type. We assume the existence of a $T$-periodic limit cycle $x_0$ for the unperturbed autonomous system. We also assume that the linearized unperturbed equation around the limit cycle has the characteristic multiplier $1$ of geometric multiplicity $1$ and algebraic multiplicity greater than $1$. The paper deals with the existence of a branch of $T$-periodic solutions emanating from the limit cycle. The problem of finding such a branch is converted into the problem of finding a branch of zeros of a suitably defined bifurcation equation $P(x,\varepsilon) +\varepsilon Q(x, \varepsilon)=0$. The main task of the paper is to define a novel equivalent integral operator having the property that the $T$-periodic adjoint Floquet solutions of the unperturbed linearized operator correspond to those of the equation $P'(x_0(\theta),0)=0$, $\theta\in[0,T]$. Once this is done it is possible to express the condition for the existence of a branch of zeros for the bifurcation equation in terms of a multidimensional Malkin bifurcation function.

The aim of this paper is to study the existence of solutions in the sense of distributions for a strongly nonlinear elliptic problem where the second term of the equation $f$ is in $ W^{-1,\overrightarrow{p}'(\,\cdot\,)}(\Omega)$ which is the dual space of the anisotropic Sobolev $W_{0}^{1,\overrightarrow{p}(\,\cdot\,)}(\Omega)$ and later $f$ will be in $L^{1}(\Omega)$.

In this paper we study functions and vector fields with isolated singularities on a $C(\mathbb{C}P^n)$-singular manifold. In general, a $C(\mathbb{C}P^n)$-singular manifold is obtained from a smooth $(2n+1)$-manifold with boundary which is a disjoint union of complex projective spaces $\mathbb{C}P^n \cup\ldots \cup\mathbb{C}P^n$ and subsequent capture of the cone over each component $\mathbb{C}P^n$ of the boundary. We calculate the Euler characteristic of a compact $C(\mathbb{C}P^n)$-singular manifold $M^{2n+1}$ with finite isolated singular points. We also prove a version of the Poincaré-Hopf Index Theorem for an almost smooth vector field with finite number of zeros on a $C(\mathbb{C}P^n)$-singular manifold.

We study an elliptic system coupled by Monge-Ampère equations: $$ \begin{cases} \det D^{2}u_{1}={(-u_{2})}^\alpha & \hbox{in $\Omega,$} \\ \det D^{2}u_{2}={(-u_{1})}^\beta & \hbox{in $\Omega,$} \\ u_{1}\le 0,\ u_{2}\le 0 & \hbox{in $\Omega,$}\\ u_{1}=u_{2}=0 & \hbox{on $ \partial \Omega,$} \end{cases} $$ here $\Omega$ is a smooth, bounded and strictly convex domain in $\mathbb{R}^{N}$, $N\geq2$, $\alpha \ge 0$, $\beta \ge 0$. When $\Omega$ is the unit ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on $\alpha$, $\beta$. When $\alpha\ge 0$, $\beta\ge 0$ and $\alpha\beta=N^2$ we also study a corresponding eigenvalue problem in more general domains.

The aim of the paper is to show the existence of positive solutions to the elliptic system of partial differential equations involving the $p$-Laplace operator \[ \begin{cases} -\Delta_p u_i(x) = f_i(u_1 (x),u_2(x),\ldots,u_m(x)), & x\in \Omega,\ 1\leq i\leq m, \\ u_i(x)\geq 0, & x\in \Omega,\ 1\leq i\leq m,\\ u(x) = 0, & x\in \partial \Omega. \end{cases} \] We consider the case of nonpositive right-hand side $f_i$, $i=1,\ldots,m$. The sufficient conditions entail spectral bounds of the matrices associated with $f=(f_1,\ldots,f_m)$. We employ the degree theory from [A. Ćwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations 254 (2013), no. 3, 1120-1136] for tangent perturbations of maximal monotone operators in Banach spaces.

We study the following nonlinear Schrödinger equation \begin{equation*} \begin{cases} -\Delta u + V(x) u = g(x,u) & \hbox{for } x\in\mathbb R^N,\\ u(x)\to 0 & \hbox{as } |x|\to\infty, \end{cases} \end{equation*} where $V\colon \mathbb R^N\to\mathbb R$ and $g\colon \mathbb R^N\times\mathbb R\to\mathbb R$ are periodic in $x$. We assume that $0$ is a right boundary point of the essential spectrum of $-\Delta+V$. The superlinear and subcritical term $g$ satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover, we get infinitely many geometrically distinct solutions provided that $g$ is odd.

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them! In fact we construct an uncountable number of infinite countable abelian groups which do have the $R_{\infty}$ property. We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group.

This paper is concerned with the following nonlinear Maxwell-Dirac system \begin{equation*} \begin{cases} \displaystyle -i\sum^{3}_{k=1}\alpha_{k}\partial_{k}u + a\beta u + \omega u-\phi u =F_{u}(x,u), \\ -\Delta \phi=4\pi|u|^{2},\\ \end{cases} \end{equation*} for $x\in\mathbb R^{3}$. The Dirac operator is unbounded from below and above, so the associated energy functional is strongly indefinite. We use the linking and concentration compactness arguments to establish the existence of ground state solutions for this system with asymptotically quadratic nonlinearity.

Let $X=\Sigma_2$ and let $F\colon X\times \mathbb{S}^1\to X\times \mathbb{S}^1$ be a map given by \[ F(x,t)=(\sigma(x),R_{x_0}(t)), \] where $(\Sigma_2,\sigma)$ denotes the full shift over the alphabet $\{0,1\}$ while $R_0$, $R_1$ are the rotations of the unit circle $\mathbb{S}^1$ by the angles $r_0$ and $r_1$, respectivelly. It was recently proved by X. Wu and G. Chen that if $r_0$ and $r_1$ are irrational, then the system $(X\times \mathbb{S}^1,F)$ has an uncountable distributionally $\delta$-scrambled set $S_\delta$ for every positive $\delta\leq \textrm{diam } X\times \mathbb{S}^1=1$. Moreover, each point in $S_\delta$ is recurrent but not weakly almost periodic (this answeres a question from [Wang et al., Ann. Polon. Math. 82 (2003), 265-272]). We generalize the above result by proving that if $r_0-r_1\in \mathbb R\setminus \mathbb Q$ and $X\subset \Sigma_2$ is a nontrivial subshift with the specification property, then the system $(X\times \mathbb{S}^1,F)$ also has the specification property. As a consequence, there exist a constant $\delta\ge 0$ and a dense Mycielski distributionally $\delta$-scrambled set for $(X\times \mathbb{S}^1,F)$, in which each point is recurrent but not weakly almost periodic.

In this paper we study the existence of nontrivial classical solution for the quasilinear Schrödinger equation: $$ - \Delta u +V(x)u+\frac{\kappa}{2}\Delta (u^{2})u= f(u), $$ in $\mathbb{R}^N$, where $N\geq 3$, $f$ has subcritical growth and $V$ is a nonnegative potential. For this purpose, we use variational methods combined with perturbation arguments, penalization technics of Del Pino and Felmer and Moser iteration. As a main novelty with respect to some previous results, in our work we are able to deal with the case $\kappa \ge 0$ and the potential can vanish at infinity.

The nonhomogeneous boundary value problem for the steady Navier-Stokes system is studied in a domain $\Omega$ with two layer type and one paraboloidal outlets to infinity. The boundary $\partial\Omega$ is multiply connected and consists of the outer boundary $S$ and the inner boundary $\Gamma$. The boundary value ${a}$ is assumed to have a compact support. The flux of ${a}$ over the inner boundary $\Gamma$ is supposed to be sufficiently small. We do not impose any restrictions on fluxes of ${a}$ over the unbounded components of the outer boundary $S$. The existence of at least one weak solution is proved.

In this work, we prove the existence of least energy nodal solutions for a class of elliptic problem in both cases, bounded and unbounded domain, when the nonlinearity has exponential critical growth in $\mathbb{R}^2$. Moreover, we also prove a nonexistence result of least energy nodal solution for the autonomous case in whole $\mathbb{R}^{2}$.

In this paper, we prove the existence of pullback attractor, pullback exponential attractor and uniform attractor for second order non-autonomous lattice system with dispersive term and time-dependent forces. Then we prove the existence of uniform exponential attractor for the system driven by quasi-periodic external forces.

One of the novelty of this paper is the study of a general class of impulsive differential equations, which is more reasonable to show dynamics of evolution processes in Pharmacotherapy. This fact reduces many difficulties in applying analysis methods and techniques in Bielecki's normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. The other novelties of this paper are new concepts of Ulam's type stability and Ulam-Hyers-Rassias stability results on compact and unbounded intervals.

In this paper, we first prove the well-posedness for the non-autonomous reaction-diffusion equations on the entire space $\mathbb{R}^N$ in the setting of locally uniform spaces with singular initial data. Then we study the asymptotic behavior of solutions of such equation and show the existence of $(H^{1,q}_U(\mathbb R^N),H^{1,q}_\phi(\mathbb R^N))$-uniform (w.r.t. $g\in\mathcal{H}_{L^q_U(\mathbb R^N)}(g_0)$) attractor $\mathcal{A}_{\mathcal{H}_{L^q_U(\mathbb R^N)}(g_0)}$ with locally uniform external forces being translation uniform bounded but not translation compact in $L_b^p(\mathbb R;L^q_U(\mathbb R^N))$. We also obtain the uniform attracting property in the stronger topology.

In this paper we study the nonlinear Schrödinger-Poisson system with a perturbation: \begin{equation*} \begin{cases} -\Delta u+u+K( x) \phi u=\vert u\vert ^{p-2}u+\lambda f(x)\vert u\vert ^{q-2}u & \text{in }\mathbb{R}^{3}, \\ -\Delta \phi =K( x) u^{2} & \text{in }\mathbb{R}^{3}, \end{cases} \end{equation*} where $K$ and $f$ are nonnegative functions, $2\le q\leq p\le 6$ and $p\ge 4$, and the parameter $\lambda \in \mathbb{R}$. Under some suitable assumptions on $K $ and $f$, the criteria of existence and multiplicity of positive solutions are established by means of the Lusternik-Schnirelmann category and minimax method.

Consider a bounded domain $\Omega\subseteq \mathbb R^3$ with smooth boundary $\partial\Omega$, a time interval $[0,T)$, $0\le T\le \infty$, and in $[0,T) \times\Omega$ the completely nonhomogeneous Navier-Stokes system $u_t - \Delta u+u\cdot \nabla u + \nabla p = f$, $u|_t=0=v_0$, ${\rm div}\,u=k$, $u|_\partial\Omega = g$, with sufficiently smooth data $f,v_0,k,g$. In this general case there are mainly known two classes of weak solutions, the class of global weak solutions, similar as in the well known case $k=0$, $g=0$ which need not be unique, see [R. Farwig, H. Kozono and H. Sohr, Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence, Rend. Sem. Math. Univ. Padova 125 (2011), 51-70], and the class of local very weak solutions, see [H. Amann, Nonhomogeneous Navier-Stokes Equations with Integrable Low-regularity Data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York, 2002, 1-26], [H. Amann, Navier-Stokes equations with nonhomogenous Dirichlet data, J. Nonlinear Math. Phys. 10, Suppl. 1 (2003), 1-11], [R. Farwig, G.P. Galdi and H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data, J. Math. Fluid Mech. 8 (2006), 423-444], which are uniquely determined but have no differentiability properties and need not satisfy an energy inequality. Our aim is to introduce the new class of local strong solutions in the usual sense for $k\not= 0$, $g\not=0$ satisfying similar regularity and uniqueness properties as in the well known case $k=0$, $g=0$. Further, we obtain precise information through the given data on the interval of existence $[0,T^*)$, $0\le T^*\le T$. Our proof is essentially based on a detailed analysis of the corresponding linear system.

Given a finite subset $\Sigma \subset \mathbb{R}$ and a positive real number $q\le 1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma ;q)=\bigg\{\sum\limits_{n=0}^{\infty }a_{n}q^{n}:(a_{n})_{n\in \omega }\in \Sigma ^{\omega }\bigg\}$, which is the unique compact solution of the equation $K=\Sigma +qK$. The obtained results are applied to studying partial sumsets $E(x)=\bigg\{\sum\limits_{n=0}^{\infty }x_{n}\varepsilon _{n}:(\varepsilon _{n})_{n\in \omega }\in \{0,1\}^{\omega } \bigg\}$ of multigeometric sequences $x=(x_{n})_{n\in \omega }$. Such sets were investigated by Ferens, Morán, Jones and others. The aim of the paper is to unify and deepen existing piecemeal results.

In this paper we study the Hutchinson-Barnsley theory of fractals in the setting of multimetric spaces (which are sets endowed with point separating families of pseudometrics) and in the setting of topological spaces. We find natural connections between these two approaches.

We introduce a class of abstract differential equation with non instantaneous impulses which extend and generalize some recent models considered in the literature. We study the existence of mild and classical solution and present some applications involving partial differential equations with non-instantaneous impulses.

We construct a $2$-dimensional torus $\mathcal{T} \subseteq \mathbb{R}^3$ having the property that it cannot be an attractor for any homeomorphism of $\mathbb{R}^3$. To this end we show that the fundamental group of the complement of an attractor has certain finite generation property that the complement of $\mathcal{T}$ does not have.

We present a result about multiplicity of solutions of asymptotically linear Dirichlet problems associated to second order equations in $\mathbb R^{2n+1}$, $n \ge 1$. Under an additional technical condition, the number of solutions obtained is given by the gap between the Morse indexes of the linearizations at zero and infinity. The additional condition is stable with respect to small perturbations of the vector field. We show with a simple example that in some cases the size of the perturbation can be explicitly estimated.

We show that the critical problem \[ -\Delta u=|u|^{{{4}}/({{N-2}})}u\quad \text{in }\Omega,\qquad\ u=0\quad \text{on }\partial\Omega, \] has at least \[ \max\{\text{cat}(\Theta,\Theta\setminus B_{r}M),\text{cupl}(\Theta ,\Theta\setminus B_{r}M)+1\}\geq2 \] pairs of nontrivial solutions in every domain $\Omega$ obtained by deleting from a given bounded smooth domain $\Theta\subset\mathbb{R}^{N}$ a thin enough tubular neighborhood $B_{r}M$ of a closed smooth submanifold $M$ of $\Theta$ of dimension $\leq N-2$, where ``cat'' is the Lusternik-Schnirelmann category and ``cupl'' is the cup-length of the pair.