Asymptotic representations of solutions of nonlinear two term difference equations

Abstract

In this paper we establish the necessary and sufficient conditions for existence of $P(\lambda )$ solutions of the nonlinear two term difference equation ${\mathrm{\Delta }}^2{y}_n=p_n\phi (y_{n+s_0})\psi (\mathrm{\Delta }{y}_{n+s_1})sgn(y_{n+s_0}),$ where $\phi$ and $\psi$ are regularly varying functions, $s_0,s_1\in \mathbb{Z}.$ Also we obtain their asymptotic representations as $n\to +\infty .$ Since $\phi$ and $\psi$ could be more complicated than power functions and $s_0,s_1\in \mathbb{Z},$ for proving the main result we formulate and prove an additional statement about existence of vanishing solutions of quasilinear system of difference equations.

Keywords:

• Emden–Fowler difference equation
• regularly varying function
• asymptotic representation
• quasi-linear system

AMS Subject Classifications:

• 39A10
• 39A12
• 39A22
• 65Q10

Notes

No potential conflict of interest was reported by the authors.