### A base-point-free definition of the Lefschetz invariant.

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We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.

Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.

We prove that a space M with Disjoint Disk Property is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. This implies that the product M × I² of a space M with the disk is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. The proof of these theorems exploits the homological characterization of Q-manifolds due to Daverman and Walsh, combined with the existence of G-stable points in C-spaces. To establish the existence of such points we prove (and afterward...

A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: ${I}_{fg}={\lambda}_{fg}$, that is, the coincidence index is equal to the Lefschetz number. It follows that if ${\lambda}_{fg}\ne 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “point-like” (acyclic)...

We prove an existence and multiplicity result for solutions of a nonlinear Urysohn type equation (2.14) by use of the Nielsen and degree theory in an annulus in the function space.