We propose an extension of the concept of Fučík spectrum to the case of coupled systems of two elliptic equations, we study its structure and some applications. We show that near a simple eigenvalue of the system, the Fučík spectrum consists (after a suitable reparametrization) of two (maybe coincident) $2$-dimensional surfaces. Furthermore, by variational methods, parts of the Fučík spectrum which lie far away from the diagonal (i.e. from the eigenvalues) are found. As application, some existence, non-existence and multiplicity results to systems with eigenvalue crossing ("jumping") nonlinearities are proved.

Lagrangian systems constrained on the closure of an open subset with Lipschitz boundary in a manifold are considered. Under suitable assumptions, the existence of infinitely many periodic solutions is proved.

A guiding function method for a class of variational inequalities is developed.

We study periodic problems driven by the scalar $p$-Laplacian with a nonsmooth potential. Using the nonsmooth critical point theory for locally Lipschitz functions, we prove two existence theorems under conditions of resonance at infinity with respect to the first two eigenvalues of the negative scalar $p$-Laplacian with periodic boundary conditions.

The purpose of this paper is to prove, under some assumptions on $g$, that the boundary value problem \begin{gather*} u'= -g(t, u, v)v, \quad v'= g(t, u, v)u, \\ u(0)=0=u(\pi), \end{gather*} has infinitely many solutions. To prove our first main result we use a theorem of J. R. Ward and to prove the second one we use Capietto-Mawhin-Zanolin continuation theorem.

Let ${\rm Homeo}(\Omega)$ be the group of all homeomorphisms of a Cantor set $\Omega$. We study topological properties of ${\rm Homeo}(\Omega)$ and its subsets with respect to the uniform $(\tau)$ and weak $(\tau_w)$ topologies. The classes of odometers and periodic, aperiodic, minimal, rank 1 homeomorphisms are considered and the closures of those classes in $\tau$ and $\tau_w$ are found.

The paper is devoted to the study of topologies on the group ${\rm Aut}(X,{\mathcal B})$ of all Borel automorphisms of a standard Borel space $(X, {\mathcal B})$. Several topologies are introduced and all possible relations between them are found. One of these topologies, $\tau$, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of ${\rm Aut}(X,{\mathcal B})$ and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in ${\rm Aut}(X,{\mathcal B})$ (Rokhlin lemma) with respect to $\tau$. It is shown that the $\tau$-closure of odometers (and of rank $1$ Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism $T\in {\rm Aut}(X,{\mathcal B})$, the concept of a Borel-Bratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism $T$ is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered Borel-Bratteli diagram. Several applications of this result are given.

A class of superlinear system of integral equations depending on multi parameters is considered. It is shown that there are three mutually exclusive and exhaustive subsets $\Theta_{1},\Gamma $ and $\Theta_{2}$ of the parameter space such that there exist at least two positive periodic solutions associated with elements in $\Theta_{1}$, at least one positive periodic solution associated with $\Gamma $ and none associated with $\Theta_{2}$.

An existence principle is established for a boundary value problem on the half line using a new theory based on weakly-strongly sequentially continuous maps.