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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Abstract

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

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

Dates
First available in Project Euclid: 25 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1337950500

Digital Object Identifier
doi:10.35834/mjms/1337950500

Mathematical Reviews number (MathSciNet)
MR2977131

Zentralblatt MATH identifier
1333.39012

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

Citation

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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