Existence of nonoscillatory solutions to second-order nonlinear neutral difference equations

Volume 8, Issue 5, pp 884--892 http://dx.doi.org/10.22436/jnsa.008.05.38

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Authors

Yazhou Tian - School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, P. R. China. - Qingdao Technological University, Feixian, Shandong 273400, P. R. China. Yuanli Cai - School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, P. R. China. Tongxing Li - Qingdao Technological University, Feixian, Shandong 273400, P. R. China.

Abstract

We study a class of second-order neutral delay difference equations with positive and negative coefficients \[\Delta(r_n(\Delta(x_n + px_{n-m}))) + p_nf(x_{n-k}) - q_ng(x_{n-l}) = 0, n = n_0, n_0 + 1,...,\] where \(p \in R, m; k; l; n_0 \in N, p_n; q_n; r_n \in R^+; f; g \in C(R;R)\) with \(xf(x) > 0\) and \(xg(x) > 0 (x \neq 0)\). Some sufficient conditions for the existence of a nonoscillatory solution of the studied equation expressed in terms of \(\sum^\infty R_np_n < 1\) and \(\sum^\infty R_nq_n < 1\) are obtained, where\(R_n = \sum^n _{s=n_0} \frac{1 }{r_s} ; n \geq n_0\).

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Keywords

• Nonoscillatory solution
• neutral delay difference equation
• second-order
• positive and negative coefficients.

MSC

•  34B15
•  34B25

References

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