DECOMPOSITION OF Cm THROUGH Q-PERIODIC DISCRETE EVOLUTION FAMILY

Abstract

Let U={U (m,n) : m,n ∈ Z+} n≥m≥0 be the q-periodic discrete evolution family of square size matrices of order m having complex scalars as entries generated by L(C^m-valued, q-periodic sequence of square size matrices (An)n∈Z+ where q≥2 is a natural number. Where the Poincare map U(q,0) is the generator of the discrete evolution family U. The main objective of this article to decompose C^m with the help of discrete evolution family. In fact we decompose Cm in two sub spaces X1 and X2 such that X1 is due to the stability of the discrete evolution family and the vectors of X1 will called stable vectors. While X2 is due to the un-stability of discrete evolution family, and their vectors will be called unstable vectors. More precisely we take the dichotomy of the discrete evolution family with the help of projection P on Cm and we discuss different results of the spaces X1 and X2 .

Keywords

• Exponential Stability
• Strong Stability
• Exponential Dichotomy
• Discrete Evolution Family