Asymptotic formulae for solutions of linear second-order difference equations

Abstract

We study asymptotic behaviour of solutions to the (nonoscillatory) linear difference equation $\Delta (r_k\Delta y_k)=p_k{y}_{k+1}$, where $p,r$ are positive sequences defined on $\{m,m+1,m+2,\dots \}\text{⊂}\mathbb{Z}$. We establish sufficient conditions (in terms of regular variation) for all eventually positive solutions to satisfy certain asymptotic formulae. As a by-product, we obtain regular variation of all these solutions and some other of their properties. Various related problems are discussed and several examples are given.

Keywords:

• linear difference equation
• asymptotic behaviour
• nonoscillatory solution
• regularly varying sequence

AMS Subject Classification:

• 39A22
• 39A12
• 26A12

Acknowledgements

The author thanks the both anonymous referees for their careful reading of the manuscript and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by RVO 67985840.