A relative homological linking of pairs is proposed. It is shown to imply homotopical linking, as well as earlier non-relative notion of homological linkings. Using Morse theory we prove a simple "homological linking principle", thereby generalizing and simplifying many well known results in critical point theory.

In this paper we discuss some problems about critical point theory. In the first part of the paper we study existence and multiplicity results of semilinear second order elliptic equation: $$ \begin{cases} -\Delta u=f(x,u) &\text{for } x\in \Omega, \\ u=0 &\text{for } x\in \partial \Omega, \end{cases} $$ In [Z. Li. Liu and S. J. Li, Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications, NoDEA 10 (2003), 133–170], the authors study the contractibility of level sets of functionals associated with some elliptic boundary value problems. In this paper by using Morse theory and minimax method we give a more precise description of topological construction of level set of critical value of energy functional for mountain pass type critical point. It is well known that nondegenerate critical point is isolated, so if a critical point is not isolated, it must be a degenerate critical point. In the second part we will give an example that all the critical points of functional of a class of oscillating equation with Neumann boundary condition are isolated and the equation has only constant solutions. Moreover, critical groups of each critical point of the functional are trivial. The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modelling the Josephson junction in superconductivity, which is of contemporary interest to physicists. The problem is similar to the elliptic sine-Gordon equation so we believe that it derives from profound physical backdrop.

In this paper we study the following $p(x)$-Laplacian problem: $$ \begin{alignat}{2} -{\rm div}(a(x)|\nabla u|^{p(x)-2}\nabla u)+b(x)|u|^{p(x)-2}u&=f(x,u) &\quad& x\in \Omega,\\ u&=0 &\quad&\text{on }\partial\Omega, \end{alignat} $$ where $1< p_{1}\le p(x)\le p_{2}< n$, $\Omega\subset {\mathbb R}^{n}$ is an exterior domain. Applying Mountain Pass Theorem we obtain the existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the $p(x)$-Laplacian problem in the superlinear case.

We prove existence of global attractors for parabolic equations of the form $$ \begin{alignedat}{2} u_t+\beta(x)u- \sum_{ij}\partial_i(a_{ij}(x)\partial_j u)&=f(x,u),&\quad &x\in \Omega,\ t\in[0,\infty[,\\ u(x,t)&=0,&\quad &x\in \partial \Omega,\ t\in[0,\infty[, \end{alignedat} $$ on an arbitrary unbounded domain $\Omega$ in $\mathbb R^3$, without smoothness assumptions on $a_{ij}(\cdot)$ and $\partial\Omega$.

In this paper we propose an elementary topological approach which unifies and extends various different results concerning fixed points and periodic points for maps defined on sets homeomorphic to rectangles embedded in euclidean spaces. We also investigate the associated discrete semidynamical systems in view of detecting the presence of chaotic-like dynamics.

Ergodic Gaussian cocycles for rigid Gaussian actions are constructed. It is also shown when any isomorphism between Gaussian actions is Gaussian.

The aim of the following paper is to propose and investigate the partial generalization of hyperbolicity to metric spaces. The locally expanding mappings, as we call them, possess many similar behaviour to that characteristic to hyperbolic mappings, in particular, they have lipschitz shadowing property. As a direct corollary we obtain for example shadowing on the Julia set.

It is shown that the set of fixed points of any $k$-uniformly lipschitzian mapping in a uniformly convex space is a retract of a domain if $k$ is close to $1$.

We study multiplicity of elastic bounce trajectories (e.b.t.'s) with fixed end points $A$ and $B$ on a nonconvex "billiard table" $\Omega$. As well known, in general, such trajectories might not exist at all. Assuming the existence of a "bounce free" trajectory $\gamma_0$ in $\Omega$ joining $A$ and $B$ we prove the existence of multiple families of e.b.t.'s $\gamma_{\lambda}$ bifurcating from $\gamma_0$ as a suitable parameter $\lambda$ varies. Here $\lambda$ appears in the dynamics equation as a multiplier of the potential term.

We use a variational approach and look for solutions as the critical points of the standard Lagrange integrals on the space $X(A,B)$ of curves joining $A$ and $B$. Moreover, we adopt an approximation scheme to obtain the elastic response of the walls as the limit of a sequence of repulsive potentials fields which vanish inside $\Omega$ and get stronger and stronger outside. To overcome the inherent difficulty of distinct solutions for the approximating problems covering to a single solutions to the limit one, we use the notion of "asymptotically critical points" (a.c.p.'s) for a sequence of functional. Such a notion behaves much better than the simpler one of "limit of critical points" and allows to prove multliplicity theorems in a quite natural way.

A remarkable feature of this framework is that, to obtain the e.b.t.'s as a.c.p.'s for the approximating Lagrange integrals, we are lead to consider the $L^2$ metric on $X(A,B)$. So we need to introduce a nonsmooth version of the definition of a.c.p. and prove nonsmooth versions of the multliplicity theorems, in particular of the "$\nabla$-theorems" used for the bifurcation result. To this aim we use several results from the theory of $\varphi$-convex functions.

It is proved that an asymptotically stable, $1$-dimensional, compact minimal set $A$ of a continuous flow on a locally compact, metric space $X$ is a periodic orbit, if $X$ is locally connected at every point of $A$. So, if the intrinsic topology of the region of attraction of an isolated, $1$-dimensional, compact minimal set $A$ of a continuous flow on a locally compact, metric space is locally connected at every point of $A$, then $A$ is a periodic orbit.