Asymptotic behavior of natural growth on time scales

ABSTRACT. The asymptotic behavior of x ∆ = px is explored, with specific reference given to how the graininess of the time scale affects stability. In addition we prove a Perron type theorem for dynamic equations on time scales. The theorem gives sufficient conditions for exponential asymptotic stability of a critical point of an almost linear dynamic equation. Application to the dynamic logistic equation is given. AMS (MOS) Subject Classification. 39A10. 1