Let $X$ be a metric space, $\pi$ be a local semiflow on $X$, $k\in{\mathbb N}$, $E$ be a $k$-dimensional normed real vector space and $\widetilde\pi$ be the semiflow generated by the equation $\dot y=Ly$, where $L\colon E\to E$ is a linear map whose all eigenvalues have positive real parts. We show in this paper that for every admissible isolated $\pi$-invariant set $S$ there is a well-defined isomorphism of degree $k$ from the (Alexander-Spanier)-cohomology categorial Conley-Morse index of $(\pi,S)$ to the cohomology categorial Conley-Morse index of $(\pi\times\widetilde\pi,S\times\{0\})$ such that the family of these isomorphisms commutes with cohomology index sequences. This extends previous results by Carbinatto and Rybakowski to the Alexander-Spanier-cohomology case.

We solve boundary value problems for the $p$-Laplacian using the notion of sandwich pairs.

We establish flows on normed spaces. Applying it we extend the results of Gromoll, Meyer, Morse and Palais for non-$C^{2}$ functionals.

We consider the semilinear problem $-\Delta u + \lambda u =|u|^{p-2}u + f(u)$ in $\Omega$, $u=0$ on $\partial \Omega$ where $\Omega \subset {\mathbb R}^N$ is a bounded smooth domain, $2< p< 2^*=2N/(N-2)$ and $f(t)$ behaves like $t^{p-1-\varepsilon}$ at infinity. We show that if $\Omega$ is invariant by a nontrivial orthogonal involution then, for $\lambda> 0$ sufficiently large, the equivariant topology of $\Omega$ is related with the number of solutions which change sign exactly once. The results are proved by using equivariant Lusternik-Schnirelmann theory.

We describe a very general procedure how one may extend an arbitrary degree or index theory (originally defined only for compact maps) also for large classes of noncompact maps. We also show how one may obtain degree or index theories relative to some set. Our results even apply to the general setting when one has a combined degree and index theory for function triples. The results are applied to countably condensing perturbations of monotone maps.

In this paper we study the existence of weighted graph-approximations of $w$-carriers whose values satisfy a certain $w$-$UV$-property. In particular, we prove that any upper semicontinuous set-valued map with compact and acyclic values (with respect to the Čech homology with rational coefficients) from a compact ANR to an ANR admits arbitrarily close weighted graph-approximations.

We define an index of Conley type for a certain class of uppersemicontinuous multivalued dynamical systems, using techniques introduced by Mrozek, Reineck and Srzednicki [The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4171–4194] for the index over the base. We give the characterisation of the nontrivial index and present an example, proving that our index detects isolated invariant sets that are not detected by Kaczyński and Mrozek's [Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), 83–96] index.

We investigate the issue of existence of maximal solutions to the vacuum Einstein solutions for asymptotically flat spacetime. Solutions are established globally in time outside a domain of influence of a suitable large compact set, where singularities can appear. Our approach shows existence of metric coefficients which obey the following behavior: $g_{\alpha\beta}=\eta_{\alpha\beta}+O(r^{-\delta})$ for a small fixed $\delta > 0$ at infinity (where $\eta_{\alpha\beta}$ is the Minkowski metric). The system is studied in the harmonic (wavelike) gauge.