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Oscillation and nonoscillation theorems for third order quasi-adjoint difference equations. (English) Zbl 0665.39002

The author investigates rarely considered third order finite difference equations \[ (1)\quad \Delta (\Delta^ 2u_ n+p_{n+1}u_{n+1})- q_{n+2}u_{n+2}=0\quad and\quad (2)\quad \Delta^ 3v_ n- p_{n+1}\Delta v_{n+1}+q_{n+1}v_{n+1}=0, \] where \(p_ n\geq 0\), \(q_ n>0\), \(\Delta p_ n+q_{n+1}>0\), \(n\geq 1\). In 22 theorems, lemmas, and corollaries various properties (in generally related to oscillatory behavior) of the solutions of (1) or (2) are presented. For example, the author proves the existence of a solution u of (1) such that \(u_ n\Delta u_ n\Delta^ 2u_ n\neq 0\), \(u_ n>0\), \(\Delta u_ n>0\), \(\Delta^ 2u_ n>0\) or of a solution v of (2) such that \(v_ n\Delta v_ n\Delta^ 2v_ n\neq 0\), \(v_ n>0\), \(\Delta v_ n<0\), \(\Delta^ 2v_ n>0\). In Theorem 3.1 the author proves that (1) is oscillatory (has any oscillatory solution) iff (2) is oscillatory. Other equivalent conditions for (1) or (2) to be oscillatory are given in Theorems 3.3, 3.6, 3.9, 3.11. Some properties of nonoscillatory solutions of (1) and (2) are presented. Notice that several of the results have their prototypes in the continuous case. See also the author’s paper [Int. J. Math. Math. Sci. 9, 781-784 (1986; Zbl 0619.39004)].
Reviewer: J.Popenda

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis

Citations:

Zbl 0619.39004
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