Theory of rapid variation on time scales with applications to dynamic equations

Jiří Vítovec

Address: Masaryk University, Faculty of Science, Department of Mathematics and Statistics, Kotlářská 2, 611 37 Brno, Czech Republic


Abstract: 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.

AMSclassification: primary 26A12; secondary 26A99, 26E70, 34N05.

Keywords: rapidly varying function, rapidly varying sequence, Karamata function, time scale, second order dynamic equation.