On a Topological Method for the Analysis of the Asymptotic Behavior of Dynamical Systems and Processes

Abstract

Publisher Summary This chapter discusses the topological method for the analysis of the asymptotic behavior of dynamical systems and processes. Wazewski principle plays an important role in the study of ordinary differential equations. Its applicability is largely due to the fact that in a finite dimensional euclidean space, the unit sphere is not a retract of the closed unit ball. Since this is no longer true in infinite-dimensional Banach spaces the direct extension of Wazewski's principle to processes or semi dynamical systems on infinite dimensional Banach spaces has a very limited applicability. Since in finite-dimensional spaces the fact that the unit sphere is not a retract of the closed unit ball is equivalent to the fact that every continuous mapping of the unity closed convex ball has a fixed point, the main idea of this work is to develop a method based on fixed point index properties instead of retraction properties.