Hyperbolicity and solvability for linear systems on time scales

Abstract

We believe that the difference between time scale systems and ordinary differential equations is not as big as people use to think. We consider linear operators that correspond to linear dynamic systems on time scales. We study solvability of these operators in ${\mathbb L}^\infty$. For ordinary differential equations such solvability is equivalent to hyperbolicity of the considered linear system. Using this approach and transformations of the time variable, we spread the concept of hyperbolicity to time scale dynamics. We provide some analogs of well-known facts of Hyperbolic Systems Theory, e.g. the Lyapunov--Perron theorem on stable manifold.