Asymptotic behavior of solutions of second order difference equations with summable coefficients. (English) Zbl 0920.39001

Consider the second order nonlinear difference equation \[ \Delta^2y_{n-1}+ q_nf(y_n)=0,\;n=1,2,\dots, \tag{*} \] where \(\Delta\) is defined by \(\Delta y_n= y_{n+1}-y_n\), under the condition that \(\lim_{n\to\infty} \sum^n_{s=1} q_s\) exists and is finite. The purpose of this paper is to obtain sufficient and/or necessary conditions for equation (*) to have solutions which behave like the nontrivial linear function \(c_1+c_2n\) as \(n\to\infty\). The authors have in view bounded and unbounded asymptotically linear solutions. The general asymptotic behavior of solutions of the equation \[ \Delta^2y_{n-1}+ q_n| y_n|^\alpha \text{sgn} y_n=0,\;n=1,2, \dots, \] where \(\alpha\) is a positive constant is also studied.


39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations