Asymptotic representation of solutions of nonlinear differential and difference equations with regularly varying coefficients

Abstract

In this dissertation, differential equations of the fourth order, difference equation of second order and cyclic systems of difference equations of second order are considered. In particular, assuming that coefficients of fourth order differential equation of Emden-Fowler type are generalized regularly varying functions, complete information about the existence of all possible intermediate regularly varying solutions and their accurate asymptotic behavior at infinity are given. The second order difference equation of Thomas-Fermy type is discussed in the framework of discrete regular variation and its strongly increasing and strongly decreasing solutions are examined in detail. Necessary and sufficient conditions for the existence of these solutions, as well as their asymptotic representations, have been determined. The obtained results enabled the complete structure of a set of regularly varying solutions to be presented. Cyclic systems of difference equations are considered as a natural generalization of second order difference equations. A full characterization of the limit behavior of all positive solutions is established. In particular, the asymptotic behavior of intermediate, as well as strongly increasing and strongly decreasing solutions is analyzed under the assumption that coefficients of the systems are regularly varying sequences and exact asymptotic formulas are derived for all these types of solutions. Also, the conditions for the existence of all types of positive solutions have been obtained.