We study a variational bifurcation problem of Böhme-Marino type associated with nonsmooth functional. The existence of two branches of bifurcation is proved.

In this paper it is studied the Dirichlet problem associated to the planar system $z'=J\nabla F(t,z)$. We consider the situation where the Hamiltonian $F$ satisfies a superquadratic-type condition at infinity.

By means of a bifurcation argument we prove the existence of infinitely many solutions. These solutions are distinguished by the Maslov index of an associated linear system.

Sufficient conditions ensuring that the solution set of some operator equations involving a duality mapping is non-empty, compact and convex are given.

We prove existence of global attractors for semilinear damped wave equations of the form $$ \begin{alignat}{2} \eps u_{tt}+\alpha(x) u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u), &\quad &x\in \Omega,t\in[0,\infty[, \\ u(x,t)&=0,&\quad& x\in \partial \Omega,\ t\in[0,\infty[. \end{alignat} $$ on an unbounded domain $\Omega$, without smoothness assumptions on $\beta(\cdot)$, $a_{ij}(\cdot)$, $f(\cdot,u)$ and $\partial\Omega$, and $f(x,\cdot)$ having critical or subcritical growth.

In this paper we look for solutions of the equation $$ \delta d\text{\bf A}=f'(\langle\text{\mathbf{A}},\text{\bf A}\rangle) \text{\bf A}\quad \text{in }\mathbb R^{2k}, $$ where $\text{\bf A}$ is a $1$-differential form and $k\geq 2$. These solutions are critical points of a functional which is strongly indefinite because of the presence of the differential operator $\delta d$. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.

We study oscillation and concentration effects for sequences of compositions $\{ f(u^\nu)\}_{\nu\in\mathbb N}$ of $\mu$-measurable functions $u^\nu\colon \Omega\rightarrow{\mathbb R}^{m}$ where $\Omega$ is the compact subset of ${\mathbb R}^n$ and $f$ is the (possibly) discontinuous function. The limits are described in terms of Young measures which can control discontinuous functions recently introduced in [A. Kałamajska, On Young measures controlling discontinuous functions, J. Conv. Anal. 13 (2006), no. 1, 177–192].

We prove the continuation property of the Conley index over a phase space for discrete semidynamical systems.

Let $f,g\colon X\to G/K$ be maps from a closed connected orientable manifold $X$ to an orientable coset space $M=G/K$ where $G$ is a compact connected Lie group, $K$ a closed subgroup and $\dim X=\dim M$. In this paper, we show that if $L(f,g)=0$ then $N(f,g)=0$; if $L(f,g)\ne 0$ then $N(f,g)=R(f,g)$ where $L(f,g), N(f,g)$, and $R(f,g)$ denote the Lefschetz, Nielsen, and Reidemeister coincidence numbers of $f$ and $g$, respectively. When $\dim X> \dim M$, we give conditions under which $N(f,g)=0$ implies $f$ and $g$ are deformable to be coincidence free.

We introduce a notion of orientation of a Fredholm operators of nonnegative index and use it in a generalized homotopy property of the respective coincidence index.

The concepts of deterministic and Kolmogorov extensions of topological flows are introduced. We show that the class of deterministic extensions contains distal extensions and moreover that for the deterministic extensions the relative topological entropy vanishes and hence they preserve the topological entropy. On the other hand we relate the Kolmogorov extensions to the asymptotic ones and we show that the class of these extensions contains uniquely ergodic u.p.e. extensions and also the class of flows admitting an invariant relative $K$-measure with full support.

The main tool used to get these results is the relative version of the Rokhlin-Sinai theorem concerning the existence of perfect measurable partitions.