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In this paper, we consider the non-autonomous reaction-diffusion equations with hereditary effects and the nonlinear term $f$ satisfying the polynomial growth of arbitrary order $p-1$ $(p\geq2)$. The delay term may be driven by a function with very weak assumptions, namely, just measurability. We extend the asymptotic a priori estimate method (see [29]) to our problem and establish a new existence theorem for the pullback attractors in $C_{L^{p}(\Omega)}$ $(p> 2)$ (see Theorem 2.12), which generalizes the results obtained in [12].

We establish spectral convergence and Conley index continuation results for a class of singularly perturbed periodic boundary value problems.

In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by $x^{\Delta} (t) = A(t) x(t) + f(t)$ and $x^{\Delta} (t) = A(t) x(t) + f(t, x(t))$, $t \in \mathbb T$, where $\mathbb T$ is an invariant under translations time scale and $A \in \mathcal{R}(\mathbb T, \mathbb R^{n \times n})$. We give new criteria ensuring the existence of an asymptotically almost automorphic solution for both equations.

We devote this paper to classifying radial solutions of a weighted semilinear elliptic equation on the hyperbolic space. More precisely, for a weighted Lane-Emden equation on the hyperbolic space, we shall study the sign and asymptotic behavior of the radial solutions. We shall also show the existence of fast-decay sign-changing solutions to the Lane-Emden equation on the hyperbolic space.

We establish a version of the Trudinger-Moser inequality involving unbounded or decaying radial weights in weighted Sobolev spaces. In the light of this inequality and using a minimax procedure we also study existence of solutions for a class of quasilinear elliptic problems involving exponential critical growth.

We consider a general integral equation satisfying algebraic conditions in a Banach space. Using Krasnosel'skiĭ-Schaefer type method and technical assumptions, we prove an existence theorem producing a periodic solution of some nonlinear integral equation.

Using the inverse integrating factor we study the limit cycles of a class of polynomial vector fields of degree $5$.

In this paper, we study the following strongly coupled quasilinear elliptic system: $$ \begin{cases} -\Delta_{p} u+\lambda a(x)|u|^{p-2}u=\dfrac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in {\mathbb R}^{N}, \\ -\Delta_{p} v+\lambda b(x)|v|^{p-2}v=\dfrac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in {\mathbb R}^{N}, \\ u,v\in D^{1,p}(\mathbb R^{N}), \end{cases} $$ where $N\geq 3$, $\lambda> 0$ is a parameter, $p< \alpha+\beta< p^{*}:={Np}/({N-p})$. Under some suitable conditions which are given in section 1, we use variational methods to obtain both the existence and multiplicity of solutions for the system on an appropriated space when the parameter $\lambda$ is sufficiently large. Moreover, we study the asymptotic behavior of these solutions when $\lambda\rightarrow\infty$.

This paper explores an abstract Riemann-Liouville fractional evolution model with a weighted delay initial condition. We develop the resolvent technique, a generalization of semigroup method, to formulate an appropriate notion of mild solutions to this abstract system and present the topological characteristics of the corresponding solution set in a weighted space. Furthermore, in view of the topological characteristics, we analyze the approximate controllability of the abstract system without Lipschitz assumption. We end up addressing an infinite dimensional fractional delay diffusion control system and a finite dimensional fractional ordinary differential control system by utilizing our theoretical findings.

In this paper, the existence of positive periodic solutions is studied for a singular Liénard equation where the weight function has an indefinite sign. Due to the lack of a priori estimates over the set of all possible positive periodic solutions in this equation, a new method is proposed for estimating a priori bounds of positive periodic solutions. By the use of a continuation theorem of the Mawhin coincidence degree, new conditions for existence of positive periodic solutions to the equation are obtained. The main results show that the singularity of coefficient function associated to the friction term at $x=0$ may help the existence of periodic solutions.

In this paper we investigate the following nonlinear Choquard equation $$ -\Delta u =\bigg(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}\,dy\bigg)g(x,u)\quad \textrm{in}\ \mathbb{R}^N, $$ where $0< \mu< N$, $N\geq3$, $g(x,u)$ is of critical growth in the sense of the Hardy-Littlewood-Sobolev inequality and $G(x,u)=\int^u_0g(x,s)\,ds$. By applying minimax procedure and perturbation technique, we obtain the existence of infinitely many solutions.

A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in the previous work [2] for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems posed in [2] regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$. Moreover, if $K$ is locally connected and each non-empty open subset of $K$ is not flat, then for any (analytic) non-meager subset $A\subset K$ the sum $A^{+n}$ of $n$ copies of $A$ is not meager in $\mathbb R^n$ (and then the sum $A^{+2n}$ of $2n$ copies of the analytic set $A$ has non-empty interior in $\mathbb R^n$ and the set $(A-A)^{+n}$ is a neighbourhood of zero in $\mathbb R^n$). This implies that a mid-convex function $f\colon D\to\mathbb R$ defined on an open convex subset $D\subset\mathbb R^n$ is continuous if it is upper bounded on some non-flat continuum in $D$ or on a non-meager analytic subset of a locally connected nowhere flat subset of $D$.

We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).

Suppose that the inverse image of the zero vector by a continuous map $f\colon {\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call ``local inessentiality of zeros'', generalizing the notion of index zero for vector fields, the $q=n$ case. We consider the problem of constructing such an approximation $g$ and a continuous homotopy $F(x,t)$ from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists $F$ also semialgebraic. If $q=2$ and $f$ is real analytic with a locally inessential zero, then there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

In this paper we study the existence of critical points of the $C^1$ functional \begin{equation*} E(u)=\frac{a}{2}\int_{\mathbb R^N}|\nabla u|^2dx+\frac{b}{4}\bigg(\int_{\mathbb R^N}|\nabla u|^2dx\bigg)^2-\frac{1}{2p}\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^pdx \end{equation*} under the constraint \begin{equation*} S_c=\bigg\{u\in H^1(\mathbb R^N)\ \bigg\vert\ \int_{\mathbb R^N}|u|^2dx=c^2\bigg\}, \end{equation*} where $a> 0$, $b> 0$, $N\geq3$, $\alpha\in(0,N)$, $ (N+\alpha)/{N}< p< (N+\alpha)/(N-2)$ and $I_{\alpha}$ is the Riesz Potential. When $p$ belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of $E$ on $S_c$. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when $c\to 0$ and $c\to +\infty$, respectively.

In this paper we are concerned with a viscoelastic wave equation with Balakrishnan-Taylor damping and frictional damping. By using the multiplier method and some properties of convex functions, we establish general energy decay rates of the equation without imposing any growth assumption near the origin on the frictional term and strongly weakening the usual assumptions on the relaxation term. Our stability result generalizes the earlier related results.

In this article we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (-\Delta)^s u = u^{-q} + \lambda u^{{2^*_s}-1}, \quad u> 0 \quad \text{in } \Omega,\quad u = 0 \quad \mbox{in } \mathbb R^n \setminus\Omega, \tag{${\rm P}_\lambda$} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s$, $ s \in (0,1)$, $ \lambda > 0$, $q> 0$ satisfies $q(2s-1)< (2s+1)$ and $2^*_s=2n/(n-2s)$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $({\rm P}_\lambda)$ in $X_0$ when $\lambda \in (0,\Lambda)$ and no solution when $\lambda> \Lambda$, where $\Lambda> 0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $({\rm P}_\lambda)$ is in $C^\alpha(\mathbb R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.

This paper deals with classifying the dynamics of topologically Anosov plane homeomorphisms. We prove that a topologically Anosov homeomorphism $f\colon\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $\overline {f(U)} \subset {\rm Int} (U)$, and such that $$ \bigcup_{n\leq 0} f^n (U)= \mathbb{R}^2.$$ In the general case, we prove a structure theorem for the $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit).

In this paper we investigate the existence of positive solutions for the problem $$ -\mathcal{L}_{K}u+V(x)u=f(u) $$% in $\mathbb R^N$, where $-\mathcal{L}_{K}$ is an integro-differential operator with measurable kernel $K$. Under apropriate hypotheses, we prove by variational methods that this equation has a nonnegative solution.