Missouri Journal of Mathematical Sciences

Existence of Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Delay Difference Equations

Zhenyu Guo, Min Liu, and Mingming Chen

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This paper studies the existence of nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation \begin{align*} \Delta \big(a_{kn} \cdots & \Delta (a_{2n} \triangle (a_{1n} \Delta (x_n + b_n x_{n-d}) ) ) \big) \\ &{} + f(n,x_{n-r_{1n}}, x_{n-r_{2n}}, \ldots , x_{n-r_{sn}} ) = 0, \ \ n \ge n_0, \end{align*} where $n_0 \ge 0$, $n \ge 0$, $d > 0$, $k > 0$, $s > 0$ are integers, $\{ a_{in} \} _{n \ge n_0}$ ($i = 1, 2, \ldots , k)$) and $\{ b_n \} _{n \ge n_0}$ are real sequences, $f \colon \{ n : n \ge n_0 \} \times {\mathbb R}^n \to {\mathbb R}$ is a mapping and $\bigcup _{j=1}^s \{ r_{jn} \} _{n \ge n_0} \subseteq {\mathbb Z}$. By applying Krasnoselskii's Fixed Point Theorem, some sufficient conditions for the existence of nonoscillatory solutions of this equation are established and indicated through five theorems according to the range of value of the sequence $b_n$.

Article information

Missouri J. Math. Sci., Volume 24, Issue 1 (2012), 67-75.

First available in Project Euclid: 25 May 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K15
Secondary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory


Guo, Zhenyu; Liu, Min; Chen, Mingming. Existence of Nonoscillatory Solutions of Higher-Order Nonlinear Neutral Delay Difference Equations. Missouri J. Math. Sci. 24 (2012), no. 1, 67--75. doi:10.35834/mjms/1337950500. https://projecteuclid.org/euclid.mjms/1337950500

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