Let $K$ be a closed, normal cone with nonempty interior ${\rm int}(K)$ in a Banach space $X$. Let $\Sigma = \{x\in{\rm int}(K) : q(x) = 1\}$ where $q \colon {\rm int}(K)\rightarrow (0,\infty)$ is continuous and homogeneous of degree $1$ and it is usually assumed that $\Sigma$ is bounded in norm. In this framework there is a complete metric $d$, Hilbert's projective metric, defined on $\Sigma$ and a complete metric $\overline d$, Thompson's metric, defined on ${\rm int}(K)$. We study primarily maps $f\colon \Sigma\rightarrow\Sigma$ which are nonexpansive with respect to $d$, but also maps $g \colon {\rm int}(K)\rightarrow {\rm int}(K)$ which are nonexpansive with respect to $\overline{d}$. We prove under essentially minimal compactness assumptions, fixed point theorems for $f$ and $g$. We generalize to infinite dimensions results of A. F. Beardon (see also A. Karlsson and G. Noskov) concerning the behaviour of Hilbert's projective metric near $\partial\Sigma := \overline\Sigma \setminus \Sigma$. If $x \in \Sigma$, $f \colon \Sigma\rightarrow \Sigma$ is nonexpansive with respect to Hilbert's projective metric, $f$ has no fixed points on $\Sigma$ and $f$ satisfies certain mild compactness assumptions, we prove that $\omega (x;f)$, the omega limit set of $x$ under $f$ in the norm topology, is contained in $\partial\Sigma$; and there exists $\eta\in\partial\Sigma$, $\eta$ independent of $x$, such that $(1 - t) y + t\eta \in\partial K$ for $0 \leq t \leq 1$ and all $y\in \omega (x;f)$. This generalizes results of Beardon and of Karlsson and Noskov. We give some evidence for the conjecture that $\text{\rm co}(\omega(x;f))$, the convex hull of $\omega(x;f)$, is contained in $\partial K$.

We study a system of (nonlinear) Schrödinger and Maxwell equation in a bounded domain, with a Dirichelet boundary condition for the wave function $\psi$ and a nonhomogeneous Neumann datum for the electric potential $\phi$. Under a suitable compatibility condition, we establish the existence of infinitely many static solutions $\psi=u(x)$ in equilibrium with a purely electrostatic field ${\mathbf{E}}=-\nabla\phi$. Due to the Neumann condition, the same electric field is in equilibrium with stationary solutions $\psi=e^{-i\omega t}u(x)$ of arbitrary frequency $\omega$.

Using variational methods we establish existence and multiplicity of positive solutions for the following class of quasilinear problems $$ -\Delta_{p}u + \lambda V(x)|u|^{p-2}u= \mu |u|^{p-2}u+|u|^{p^{*}-2}u \quad\text{in } {\mathbb R}^{N} $$ where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$, $p^{*}={pN}/(N-p)$, $\lambda, \mu \in (0, \infty)$ and $V\colon {\mathbb R}^{N} \rightarrow {\mathbb R}$ is a continuous function verifying some hypothesis.

The existence and concentration behavior of nodal solutions are established for the equation $-\varepsilon^{p} \Delta_{p}u + V(z)|u|^{p-2}u=f(u)$ in $\Omega$, where $\Omega$ is a domain in ${\mathbb R}^{N}$, not necessarily bounded, $V$ is a positive Hölder continuous function and $f\in C^{1}$ is a function having subcritical growth.

Long time existence of axially symmetric solutions to the Navier-Stokes equations in a bounded cylinder and with boundary slip conditions is proved. The axially symmetric solutions with nonvanishing azimuthal component of velocity (swirl) are examined. The solutions are such that swirl is small in a neighbourhood close to the axis of symmetry but it is large in some positive distance from it. There is a great difference between the proofs of global axially symmetric solutions with vanishing and nonvanishing swirl. In the first case global estimate follows at once but in the second case we need a lot of considerations in weighted spaces to show it.

The existence is proved by the Leray-Schauder fixed point theorem.

The nonstationary Navier-Stokes problem is studied in a three-dimensional domain with cylindrical outlets to infinity in weighted Sobolev function spaces. The unique solvability of this problem is proved under natural compatibility conditions either for a small time interval or for small data. Moreover, it is shown that the solution having prescribed fluxes over cross-sections of outlets to infinity tends in each outlet to the corresponding time-dependent Poiseuille flow.

A system of quasilinear non-uniformly parabolic-elliptic equations modelling chemotaxis and taking into account the volume filling effect is studied under no-flux boundary conditions. The proof of existence and uniqueness of a global-in-time weak solution is given. First the local solutions are constructed. This is done by the Schauder fixed point theorem. Uniqueness is proved with the use of the duality method. A priori estimates are stated either in the case when the Lyapunov functional is bounded from below or chemotactic forces are suitably weakened.

A criterion for existence of a fixed point for an affine action of a given group on a compact convex space is presented. From this we derive that a discrete countable group is amenable if and only if there exists an invariant probability measure for any action of the group on a Hilbert cube. Amenable properties of the group of all isometries of the Urysohn universal homogeneous metric space are also discussed.