A few remarks on Poincaré-Perron solutions and regularly varying solutions

Abstract

Abstract We establish conditions guaranteeing the existence of (generalized) regularly varying solutions to nth order linear differential and q-difference equations. The proofs are based mainly on classical Poincaré’s and Perron’s theorems and certain transformations. In some special cases, our results reduce to the existing ones, thus actually we offer an alternative approach to some parts of the asymptotic theory of differential equations made in the framework of regular variation. For higher order cases, our statements are essentially new. Another important feature of this paper is that it reveals connections among various results and somehow revises some of them.