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Existence of a nonoscillatory solution of a second-order linear neutral difference equation. (English) Zbl 1144.39004
The author considers the neutral delay difference equation with positive and negative coefficients $$\Delta^2(x_n+p x_{n-m})+p_nx_{n-k}-q_n x_{n-\ell}=0\tag1$$ where $p\in\Bbb R$ and $m,k,\ell\in\Bbb N$ and $p_n,q_n\in\Bbb R^+$, $n\ge n_0\in\Bbb N$. The main result is the following Theorem: If the conditions $$\sum^\infty ip_i<\infty,\quad \sum^\infty iq_i<\infty$$ hold, where $p\ne -1$, then equation (1) has a nonoscillatory solution.

39A11Stability of difference equations (MSC2000)
39A10Additive difference equations
Full Text: DOI
[1] Ladas, G.: Recent developments in the oscillations of delay difference equations. Differential equations: stability and control (1990) · Zbl 0731.39002
[2] Lalli, B. S.; Zhang, B. G.; Zhao, L. J.: On the oscillations and existence of positive solutions of neutral difference equations. J. math. Anal. appl. 158, 213-233 (1991) · Zbl 0732.39002
[3] Lalli, B. S.; Zhang, B. G.: On existence of positive solutions and bounded oscillations for neutral difference equations. J. math. Anal. appl. 166, 272-287 (1992) · Zbl 0763.39002
[4] Shen, J. H.; Wang, Z. C.; Qian, X. Z.: On existence of positive solutions of neutral difference equations. Tamkang. J. Math. 25, 257-265 (1994) · Zbl 0809.39001
[5] Bainov, D. D.; Mishev, D. P.: Oscillation theory for neutral differential equations with delay. (1991) · Zbl 0747.34037
[6] Gyori, I.; Ladas, G.: Oscillation theory for delay differential equations with applications. (1991)
[7] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[8] Kulenovic, M. R. S.; Hadziomerspahic, S.: Existence of nonoscillatory solution of second order linear neutral delay equation. J. math. Anal. appl. 228, 436-448 (1998)
[9] Zhang, B. G.; Yu, J. S.: On existence of positive solutions for neutral differential equations. Sci. China ser. A 8, 785-790 (1992)