In this paper we define attractors and Morse decompositions in an abstract framework of curves in a metric space. We establish some basic properties of these concepts including their stability under perturbations. This extends results known for flows and semiflows on metric spaces to large classes of ordinary or partial differential equations with possibly nonunique solutions of the Cauchy problem. As an application, we first prove a Morse equation in the context of a Conley index theory which was recently defined in [M. Izydorek and K. P. Rybakowski, On the Conley index in Hilbert spaces in the absence of uniqueness, Fund. Math.] for problems without uniqueness, and then apply this equation to give an elementary proof of two multiplicity results for strongly indefinite elliptic systems previously obtained in [S. Angenent and R. van der Vorst, A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology, Math. Z. 231 (1999), 203–248] using Morse-Floer homology.

We show that a family $F_p$; $p\in P$ of nonlinear elliptic boundary value problems of index $0$ parametrized by a compact manifold admits a reduction to a family of compact vector fields parametrized by $P$ if and only if its index bundle ${\rm Ind}\,F$ vanishes. Our second conclusion is that, in the presence of bounds for the solutions of the boundary value problem, the non vanishing of the image of the index bundle under generalized $J$-homomorphism produces restrictions on the possible values of the degree of $F_p$. The most striking manifestation of this arises when the first Stiefel-Whitney class of the index bundle is nontrivial. In this case, the degree of $F_p$ must vanish! From this we obtain a number of corollaries about bifurcation from infinity for solutions of nonlinear elliptic boundary value problems.

Let $N \geq 3$ and $\Omega \subset \mathbb R^{N} $ be a bounded domain with a smooth boundary $\partial \Omega $. We consider a semilinear boundary value problem of the form $$ \begin{cases} -\Delta u = \vert u\vert ^{2^*-2} u +f &\text{\rm in } \Omega,\\ u> 0 & \text{\rm in } \Omega, \\ u= 0 & \text{\rm on } \partial \Omega, \end{cases} \leqno \text{\rm (P)} $$ where $f \in C(\overline \Omega )$ and $2^* = 2N/(N-2)$. We show the effect of topology of $\Omega $ on the multiple existence of solutions.

We prove existence and upper semicontinuity of attractors for a reaction-diffusion equation on a family of thin unbounded domains collapsing onto a lower dimensional subspace.

We consider a class of asymptotically linear variational inequalities. We show the existence of a nontrivial solution under assumptions which allow the problem to be degenerate at the origin.

We establish a variant of the symmetric quasi linear first order system given by H. Friedrich for the evolution equations of gravitating fluid bodies in General Relativity which can be important to solve realistic problems. Our version has the advantage of introducing only physical characteristics. We state explicitly the conditions under which the system is hyperbolic and admits a well posed Cauchy problem.

Existence of almost-periodic solutions to quasi-linear evolution inclusions under a Stepanov almost-periodic forcing is nontraditionally examined by means of the Banach-like and the Schauder-Tikhonov-like fixed-point theorems. These multivalued fixed-point principles concern condensing operators in almost-periodic function spaces or their suitable closed subsets. The Bohr-Neugebauer-type theorem jointly with the Bochner transform are employed, besides another, for this purpose. Obstructions related to possible generalizations are discussed.

Given an $(m-1)$-dimensional cycle $z$ in a finitely generated acyclic chain complex, we want to explicitly construct an $m$-dimensional chain ${\rm Cob}(z)$ whose algebraic boundary is $z$. The acyclicity of the chain complex implies that a solution exists (it is not unique) but the traditional linear algebra methods of finding it lead to a high complexity of computation. We are searching for more efficient algorithms based on geometric considerations. The main motivation for studying this problem comes from the topological and computational dynamics, namely, from designing general algorithms computing the homomorphism induced in homology by a continuous map. This, for turn, is an essential step in computing such invariants of dynamical properties of nonlinear systems as Conley index or Lefschetz number. Another potential motivation is in the relationship of our problem to the problem of finding minimal surfaces of closed curves.

The concept of $0$-epi maps is a known homotopic analogue to maps with nonzero degree. There exist various related notions on unbounded sets and for multivalued maps. We introduce a concept which unifies these definitions. We also compare the various concepts. In particular, we prove that proper $0$-epi maps are also $0$-multiepi.

Given a nonempty closed subset $A$ of a Banach space $X$ and a point $x \in X$, we consider the problem of finding a nearest point to $x$ in $A$. We define an appropriate complete metric space $\mathcal M$ of all pairs $(A,x)$ and construct a subset $\Omega$ of $\mathcal M$ which is the countable intersection of open everywhere dense sets such that for each pair in $\Omega$ this problem is well-posed. As a matter of fact, we show that the complement of $\Omega$ is not only of the first category, but also sigma-porous.