Non-oscillation of second order linear self-adjoint nonhomogeneous difference equations.

*(English)*Zbl 1249.39015The author considers the linear second order difference equation \(\Delta (p_{n-1}\Delta y_{n-1})+q_ny_n=0\), the nonlinear nonhomogeneous second order difference equation \(\Delta (p_{n-1}\Delta y_{n-1})+q_ng(y_n)=f_{n-1}\), where \(f_n\geq 0\) or \(f_n\leq 0\) and \(ug(u)>0\) for \(u\neq 0\), and the linear third order difference equation \(y_{n+2}+a_ny_{n+1}+b_ny_n+c_ny_{n-1}=g_{n-1}\); always \(p_n>0\). Sufficient and/or necessary conditions for (non)oscillation of the linear second order equation and sufficient conditions for nonoscillation of all solutions to second order nonlinear and third order difference equations are obtained.

The paper is somehow problematic, especially the part concerning linear second order equations. In general remarks, the author does not mention the condition \(p_n\neq 0\) that enables us to include also the Fibonacci equation, which is mentioned in the text. Further, there is a discussion about lack of conditions in the literature which are sufficient and necessary for (non)oscillation of the equation. But, first, the author ignores some papers containing related results, second, he perhaps is not aware that such an effective condition in general case is practically not possible to be obtained for second order linear difference equations. The most problematic part is Section 2. Indeed, although there are several theorems which are claimed to be new, Theorem 1 is a trivial consequence of the Sturm type comparison theorem, Theorem 3 is a trivial consequence of the Leighton-Wintner type criterion, and Theorem 5 is a trivial consequence of known results. Further, the part concerning the forced equation does not contain any mention about related topics. It seems that the choice of the literature is somehow self-centered and many important works are not mentioned.

The paper is somehow problematic, especially the part concerning linear second order equations. In general remarks, the author does not mention the condition \(p_n\neq 0\) that enables us to include also the Fibonacci equation, which is mentioned in the text. Further, there is a discussion about lack of conditions in the literature which are sufficient and necessary for (non)oscillation of the equation. But, first, the author ignores some papers containing related results, second, he perhaps is not aware that such an effective condition in general case is practically not possible to be obtained for second order linear difference equations. The most problematic part is Section 2. Indeed, although there are several theorems which are claimed to be new, Theorem 1 is a trivial consequence of the Sturm type comparison theorem, Theorem 3 is a trivial consequence of the Leighton-Wintner type criterion, and Theorem 5 is a trivial consequence of known results. Further, the part concerning the forced equation does not contain any mention about related topics. It seems that the choice of the literature is somehow self-centered and many important works are not mentioned.

Reviewer: Pavel Řehák (Brno)

##### MSC:

39A21 | Oscillation theory for difference equations |

39A06 | Linear difference equations |

39A10 | Additive difference equations |