q-Karamatine funkcije i asimptotska svojstva rešenja nelinearnih q-diferencnih jednačina

Abstract

The purpose of the doctoral dissertation is to determine the conditions for the existence and to examine in detail the asymptotic properties of solutions of the second order nonlinear q-difference equations, with an application of the theory of q-regular variation. The half-linear q-difference equation was analyzed in the framework of q-regular variation. Necessary and sufficient conditions for the existence of q-regularly varying solutions of the half-linear q- difference equation were obtained. Moreover, sufficient conditions for all eventually positive solutions to be q-regularly varying were examined. In cases where this is possible, the application of q-Karamata’s integration theorem and properties of q-regularly varying functions have been used to determine the precise asymptotic formula of different types of solutions, which accurately describes the behavior of these solutions in long time intervals, which is of special importance from the point of view of application. The obtained results in the q-calculus were compared with the known results in the continuous and the discrete case, but also, they were used to obtain new results in the discrete asymptotic theory. The sublinear second order q-difference equation of Emden-Fowler type was also analyzed in the framework of q-regularly varying functions. Assuming that the coefficients of this equation are q-regularly varying functions, necessary and sufficient conditions for the existence of strongly increasing and strongly decreasing solutions, as well as their asymptotic representations at infinity, have been determined. Moreover, it was shown that all q-regularly varying solutions of the same regularity index have the same asymptotic representation at infinity. The obtained results enabled the complete structure of the set of q-regularly varying solutions to be presented.