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Oscillatory and asymptotic behavior of fourth order nonlinear delay difference equations. (English) Zbl 1009.39007
From the summary: The authors consider the nonlinear difference equation $\Delta^2 \bigl(a_n\Delta (b_n\Delta y_n)\bigr) +f(n,y_{n-l})=0, \quad n\in N(n_0)= \{n_0,n_0+1, \dots\},$ where $$\{a_n\}$$ and $$\{b_n\}$$ are positive real sequences, $$l$$ is a nonnegative integer, $$f:N(n_0)\times \mathbb{R} \to\mathbb{R}$$ is a continuous function with $$uf(n,u)>0$$ for all $$u\neq 0$$. They obtain necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior. They also obtain sufficient conditions for all solutions to be oscillatory if $$f$$ is either strongly sublinear or strongly superlinear.

##### MSC:
 39A11 Stability of difference equations (MSC2000)