It is shown that an autonomous delay differential system for a damped spring with a delayed restoring force has a periodic solution whose orbit is exponentially stable with asymptotic phase.

In this paper we shall study the semilinear elliptic problem $$ \begin{cases} -\Delta u +\sigma(x)u= |u|^{p-2}u + f(x) & \text{in }\mathbb R^N,\\ u\rightarrow 0\quad\text{as } |x| \rightarrow\infty, \end{cases} $$ where $\sigma(x) \rightarrow\infty$ as $| x| \rightarrow\infty$, $p> 2$ and $f\in L^{2}(\mathbb R^{N})$. Thanks to a compact embedding of a suitable weigthed Sobolev space in $L^{2}(\mathbb R^{N})$, a direct use of the Symmetric Mountain Pass Theorem (if $f=0$) and of the fibering method (if $f\neq 0$) allows to extend some multiplicity results, already known in the case of bounded domains.

A semilinear elliptic equation, $-\Delta u=\lambda f(u)$, is studied in a ball with the Dirichlet boundary condition. For a closed subgroup $G$ of the orthogonal group, it is proved that the number of non-radial $G$ invariant solutions diverges to infinity as $\lambda$ tends to $\infty$ if $G$ is not transitive on the unit sphere.

We consider the index of the solution set of Fredholm equations with $f$-condensing type perturbations. This characteristic is applied to the existence of periodic solutions for delay differential equations.

Is it possible to establish a new Morse theory if the function $f$ losses the (PS) condition at some isolated values? Yes, it is! In this paper we will recall a such a theory. One of the purposes of establishing such a theory is to consider multiplicity results for strong resonance problems and to deal with multiple resonant energy levels. Both of these questions were not studied much in the past because of the limitation of methods. Using the new Morse theory we can deal with these problems.

In this paper we will improve some results previously obtained, showing that the so called $\varphi$-category of a pair of manifolds is infinte uncountable under certain topological conditions on the two given manifolds.

The Nielsen coincidence theory is well understood for a pair of maps between $n$-dimensional compact manifolds for $n$ greater than or equal to three. We consider coincidence theory of a pair $(f,g)\colon K \to \mathbb N^n$, where $K$ is a finite simplicial complex of the same dimension as the manifold $\mathbb N^n$. We construct an algorithm to find the minimal number of coincidences in the homotopy class of the pair based on the obstruction to deform the pair to coincidence free. Some particular cases are analyzed including the one where the target is simply connected.

The relative Reidemeister number, denoted by ${\rm R}(f;X,A)$, is an upper bound for the relative Nielsen number, denoted by ${\rm N}(f;X,A)$. If $(f,f_{A})$ is a pair of fibre-preserving maps of a pair of Hurewicz fibrations, then under certain conditions, the relative Reidemeister number can be calculated in terms of those on the base and on the fiber. In this paper, we give addition formulas for ${\rm R}(f;X,A)$ and for the relative Reidemeister number on the complement ${\rm R}(f;X-A)$. As an application, we give estimation of the asymptotic Nielsen type number ${\rm NI}^{\infty}(f)$.

In [A. L. Fel’shtyn and R. Hill, The Reidemeister zeta function with applications to Nielsen theory and connection with Reidemeister torsion, K-Theory 8 (1994), 367–393] we have conjectured that the Reidemeister number is infinite as long as an endomorphism of a discrete group is injective and the group has exponential growth. In the paper we prove this conjecture for any automorphism of a non-elementary, Gromov hyperbolic group. We also prove some generalisations of this result. The main results of the paper have topological counterparts.

The Thom-Pontryagin construction is studied from the point of view of fixed point situations, and a very natural correspondence between framed cobordism classes and fixed point situations is given. Since fixed point classes integrate a cohomology theory, called ${\rm FIX}$, which generalizes in a natural way to an equivariant theory, this sheds light into possible approaches to equivariant cobordism.

We use the ideas of Lusternik-Schnirelmann theory to describe the set of fixed points of certain homotopy equivalences of a general space. In fact, we extend Lusternik-Schnirelmann theory to pairs $(\varphi, f)$, where $\varphi$ is a homotopy equivalence of a topological space $X$ and where $f \colon X \rightarrow \mathbb R$ is a continuous function satisfying $f(\varphi(x)) < f(x)$ unless $\varphi (x) = x$; in addition, the pair $(\varphi, f)$ is supposed to satisfy a discrete analogue of the Palais-Smale condition. In order to estimate the number of fixed points of $\varphi$ in a subset of $X$, we consider different relative categories. Moreover, the theory is carried out in an equivariant setting.