On condensing discrete dynamical systems

summary:In the paper the fundamental properties of discrete dynamical systems generated by an $\alpha$-condensing mapping ($\alpha$ is the Kuratowski measure of noncompactness) are studied. The results extend and deepen those obtained by M. A. Krasnosel'skij and A. V. Lusnikov in \cite{21}. They are also applied to study a mathematical model for spreading of an infectious disease investigated by P. Takac in \cite{35}, \cite{36}