Stability of fixed points for a class of quasilinear cascades in the space conv $R^n$

Abstract

The discrete dynamical systems (cascades) in semilinear metric space of nonempty convex compacts of finite-dimensional space are studied. Using the methods of convex geometry of H. Minkowski and A. D. Alexandrov the sufficient conditions of the stability of the fixed points were established. Under certain restrictions on the mappings generating the cascade, the problem of asymptotic stability of fixed point of the cascade was reduced to localization of the roots of a polynomial inside the unit circle in the complex plane. Examples of cascades in the plane were given.