## 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.

### MSC:

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