In this paper we treat the problem of the rate of convergence of attractors of dynamical systems for some autonomous semilinear parabolic problems. We consider a prototype problem, where the diffusion $a_0(\cdot)$ of a reaction-diffusion equation in a bounded domain $\Omega$ is perturbed to $a_\varepsilon(\cdot)$. We show that the equilibria and the local unstable manifolds of the perturbed problem are at a distance given by the order of $\|a_\varepsilon-a_0\|_\infty$. Moreover, the perturbed nonlinear semigroups are at a distance $\|a_\varepsilon-a_0\|_\infty^\theta$ with $\theta< 1$ but arbitrarily close to $1$. Nevertheless, we can only prove that the distance of attractors is of order $\|a_\varepsilon-a_0\|_\infty^\beta$ for some $\beta< 1$, which depends on some other parameters of the problem and may be significantly smaller than $1$. We also show how this technique can be applied to other more complicated problems.

We apply the so-called $p$-regularity theory to prove the existence of solutions to two nonlinear boundary value problems: an equation of rod bending and some nonlinear Laplace equation.

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation $$ \begin{alignat}{2} u_{tt}+\alpha u_t+\beta(x)u-\Delta u& =f(x,u), &\quad&(t,x)\in[0,+\infty[\times\Omega, \\ u&=0,&\quad &(t,x)\in[0,+\infty[\times\partial\Omega, \end{alignat} $$ in $H^1_0(\Omega)\times L^2(\Omega)$ has finite Hausdorff and fractal dimension. Here $\Omega$ is a regular, possibly unbounded, domain in $\mathbb R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff and fractal dimension of $\mathcal I$ in terms of the structure parameters of the equation.

We generalize an existence result on second order systems with a nonlinear term satisfying the so-called Hartman-Nagumo condition. The generalization is based on the use of Gauss second fundamental form and continuation techniques.

In this paper, we investigate the existence of infinite many positive solutions for the nonlinear first-order BVP with integral boundary conditions $$ \begin{cases} x^{\Delta}(t)+p(t)x^{\sigma}(t)=f(t,x^{\sigma}(t)), & t\in (0,T)_{\mathbb{T}}, \\ x(0)-\beta x^{\sigma}(T)=\alpha\int_{0}^{\sigma(T)}x^{\sigma}(s)\Delta g(s), \end{cases} $$ where $x^{\sigma}=x\circ\sigma$, $f\colon [0,T]_{\mathbb{T}}\times\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ is continuous, $p$ is regressive and rd-continuous, $\alpha,\beta\geq0$, $g\colon [0,T]_{\mathbb{T}}\rightarrow \mathbb{R}$ is a nondecreasing function. By using the fixed-point index theory and a new fixed point theorem in a cone, we provide sufficient conditions for the existence of infinite many positive solutions to the above boundary value problem on time scale $\mathbb{T}$.

Given a bounded domain $\Omega\subset\mathbb{R}^N$ of class $C^k$ with $k\ge3$, we prove that for a generic deformation $I+\psi$, with $\psi$ small enough, all the critical points of the mean curvature of the boundary of the domain $(I+\psi)\Omega$ are non degenerate.

We establish the robustness of nonuniform exponential dichotomies under sufficiently small analytic parameter-dependent perturbations. We also show that the stable and unstable subspaces of the exponential dichotomies depend analytically on the parameter.

We consider the Schrödinger-Poisson-Slater (SPS) system in $\mathbb R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal radially symmetric solution which changes sign exactly $k$ times in the radial variable.

Moreover, when the domain is the ball of $\mathbb R^3$ we obtain the existence of radial global solutions for the associated nonlocal parabolic problem having $k+1$ nodal regions at every time.

We discuss a class of models arising in the study of circadian rhythm and the properties of the matrix equations providing the bifurcation points for a wide parameter class. In particular, we prove that the five dimensional system studied in the cited work of Gonze, Halloy, and Goldbeter can have only a simple Hopf bifurcation.

We study constructions of vector fields with properties which are characteristic to Reeb vector fields of contact forms. In particular, we prove that all closed oriented odd-dimensional manifold have geodesible vector fields.

In this paper, we consider a concept of local Nash equilibrium for non-cooperative games - the so-called weak local Nash equilibrium. We prove its existence for a significantly more general class of sets of strategies than compact convex sets. The theorems on existence of the weak local equilibrium presented here are applications of Brouwer and Lefschetz fixed point theorems.

This paper is a systematic study about the syndetically proximal relation and the possible existence of syndetically scrambled sets for the dynamics of continuous self-maps of compact metric spaces. Especially we consider various classes of transitive subshifts, interval maps, and topologically Anosov maps. We also present many constructions and examples.