Robert Skiba

Global bifurcation of homoclinic trajectories of discrete dynamical systems

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving topological properties of the asymptotic stable bundles.

A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

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.

Editors’ preface for the topical issue “Topics in Nonlinear Analysis”

The contents of the current issue of the Central European Journal of Mathematics concerns topics discussed during the Sixth Symposium in Nonlinear Analysis, held on September 7–9, 2011 in Toruń, Poland and organized by the Nicolaus Copernicus University in Toruń, Marie Skłodowska-Curie University in Lublin and the Juliusz Schauder Center for Nonlinear Studies in Toruń. The main aim of this national conference was to bring Polish researchers working in the area of broadly understood nonlinear mathematical analysis together and to create a platform for discussions and exchange of experience between the participants. The term ‘nonlinear analysis’ is used here in the broadest possible sense, in accordance with the ideas of Henri Poincaré, one of the greatest pioneers of the subject. It contains the study of specific nonlinear problems that arise naturally in differential geometry, classical and modern mathematical physics, the calculus of variations together with special methods and techniques including general and algebraic topology, group theory, linear algebra and functional analysis.

Graph-approximation of multivalued weighted maps

In this paper we study the existence of weighted graph-approximations of

On a generalization of absolute neighborhood retracts

w

Fixed point theory of multivalued weighted maps

-carriers whose values satisfy a certain

Topology and homoclinic trajectories of discrete dynamical systems

w

A cohomological index of Fuller type for set-valued dynamical systems

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Fixed points of multivalued weighted maps

UV

On the Lefschetz fixed point theorem for multivalued weighted mappings

-property. In particular, we prove that any upper semicontinuous set-valued map with compact and acyclic values (with respect to the Cech homology with rational coefficients) from a compact ANR to an ANR admits arbitrarily close weighted graph-approximations.