Missouri Journal of Mathematical Sciences

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

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

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

References

• R. P. Agarwal, Difference Equations and Inequalities, 2nd ed., Dekker, New York, 2000.
• R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kulwer Academic, 2000.
• R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, Oscillations of higher-order neutral difference equations, Appl. Math. Lett., 10 (1997), 71–78.
• R. P. Agarwal and S. R. Grace, The oscillation of higher-order nonlinear difference equations of neutral type, Appl. Math. Lett., 12 (1999), 77–83.
• S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Anal., 20 (1993), 193–203.
• J. F. Cheng, Existence of a nonoscillatory solution of a second-order linear neutral difference equation, Appl. Math. Lett., 20 (2007), 892–899.
• I. Gyori and G. Ladas, Oscillation Theory for Delay Differential Equations with Applications, Oxford Univ. Press, London 1991.
• Z. Liu, Y. Xu, and S. M. Kang, Global solvability for a second order nonlinear neutral delay difference equation, Comput. Math. Appl., 57 (2009), 587–595.
• Q. Meng and J. Yan, Bounded oscillation for second-order nonlinear difference equations in critical and non-critical states, J. Comput. Appl. Math., 211 (2008), 156–172.
• M. Migda and J. Migda, Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Anal., 63 (2005), 789–799.
• E. Thandapani, M. M. S. Manuel, J. R. Graef, and P. W. Spikes, Monotone properties of certain classes of solutions of second-order difference equations, Comput. Math. Appl., 36 (2001), 291–297.
• F. Yang and J. Liu, Positive solution of even order nonlinear neutral diference equations with variable delay, J. Systems Sci. Math. Sci., 22 (2002), 85–89.
• B. G. Zhang and B. Yang, Oscillation of higher order linear difference equation, Chinese Ann. Math., 20 (1999), 71–80.
• Z. G. Zhang and Q. L. Li, Oscillation theorems for second-order advanced functional difference equations, Comput. Math. Appl., 36 (1998), 11–18.
• Y. Zhou, Existence of nonoscillatory solutions of higher-order neutral difference equations with general coefficients, Appl. Math. Lett., 15 (2002), 785–791.
• Y. Zhou and Y. Q. Huang, Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations, J. Math. Anal. Appl., 280 (2003), 63–76.
• Y. Zhou and B. G. Zhang, Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients, Comput. Math. Appl., 45 (2003), 991–1000.