Theory of Rapid Variation on Time Scales with Applications to Dynamic Equations

In the first part of this paper we establish the theory of rapid variation on time scales, which corresponds to existing theory from continuous and discrete cases. We introduce two definitions of rapid variation on time scales. We will study their properties and then show the relation between them. In the second part of this paper, we establish necessary and sufficient conditions for all positive solutions of the second order half-linear dynamic equations on time scales to be rapidly varying. Note that these results are new even for the linear (dynamic) case and for the half-linear discrete case. In the third part of this paper we give a complete characterization of all positive solutions of linear dynamic equations and of all positive decreasing solutions of half-linear dynamic equations with respect to their regularly or rapidly varying behavior. The paper is finished by concluding comments and open problems of these themes.