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Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations. (English) Zbl 1182.34113

From the introduction: We study the asymptotic nature of all bounded solutions of a higher-order nonlinear forced neutral dynamic equation of the following form:
\[ [x(t)+A(t)x(\alpha(t))]^{\Delta^n}+f(t,x(\beta(t)),x(\gamma(t)))=\varphi(t)\text{ for }t\in[t_0,\infty)_{\mathbb T},\tag{1} \]
where \(n\in\mathbb N\), \(t_0\in\mathbb T=\infty\), \(A\in \text{C}_{\text{rd}}(t_0,\infty)_{\mathbb T}\times \mathbb R^2,\mathbb R)\), \(\varphi\in\text{C}_{\text{rd}}[t_0,\infty)_{\mathbb T},\mathbb R)\) and \(\alpha,\beta,\gamma\in\text{C}([t_0,\infty)_{\mathbb T},\mathbb T)\) such that \(\alpha\) has the inverse \(\alpha^{-1}\in\text{C}(\mathbb T,\mathbb T)\) when needed, as well as the condition \(\lim_{t\to\infty}\alpha(t)=\lim_{t\to\infty}\beta(t)=\lim_{t\to\infty}\gamma(t)=\infty\) being satisfied.

MSC:

34N05 Dynamic equations on time scales or measure chains
34K25 Asymptotic theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K40 Neutral functional-differential equations
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[1] Agarwal, R. P., Difference Equations and Inequalities: Theory Methods and Applications (2000), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0952.39001
[2] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 1-2, 3-22 (1999) · Zbl 0927.39003
[3] Agarwal, R. P.; Bohner, M.; Řehák, P., Half-linear dynamic equations, Nonlinear Analysis and Applications, vol. 1-2 (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, pp. 1-58 · Zbl 1056.34049
[4] Agarwal, R. P.; Bohner, M.; Li, W. T., Nonoscillation and Oscillation: Theory for Functional Differential Equations (2004), Marcel Dekker
[5] Agarwal, R. P.; Bohner, M.; Grace, S. R.; O’Regan, D., Discrete Oscillation Theory (2005), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1084.39001
[6] Agarwal, R. P.; Grace, S. R.; O’Regan, D., Oscillation Theory for Difference and Functional Differential Equations (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0969.34062
[7] Akın-Bohner, E.; Bohner, M.; Djebali, S.; Moussaoui, T., On the asymptotic integration of nonlinear dynamic equations, Adv. Differ. Equat., 739602, 17 (2008) · Zbl 1149.34033
[8] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[9] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[10] Bolat, Y.; Akın, Ö., Oscillatory behaviour of higher order neutral type nonlinear forced differential equation with oscillating coefficients, J. Math. Anal. Appl., 290, 1, 302-309 (2004) · Zbl 1055.34124
[11] Erbe, L. H.; Kong, Q.; Zhang, B. G., Oscillation Theory for Functional-Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 190 (1995), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York
[12] Erbe, L. H.; Peterson, A.; Saker, S. H., Hille and Nehari type criteria for third-order dynamic equations, J. Math. Anal. Appl., 329, 112-131 (2007) · Zbl 1128.39009
[13] Győri, I.; Ladas, G., Oscillation Theory of Delay Differential Equations: With Applications (1991), Oxford Science Publications, The Clarendon Press, Oxford University Press: Oxford Science Publications, The Clarendon Press, Oxford University Press New York · Zbl 0780.34048
[14] Karpuz, B.; Padhy, L. N.; Rath, R. N., On oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients, Electron. J. Differ. Equat., 2008, 113, 1-15 (2008) · Zbl 1171.34043
[15] Karpuz, B., Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electron. J. Qual. Theory Differ. Equat., 2009, 34, 1-14 (2009) · Zbl 1184.34072
[16] Kubiaczyk, I.; Li, W. T.; Saker, S. H., Oscillation of higher order delay differential equations with applications to hyperbolic equations, Indian J. Pure Appl. Math., 34, 8, 1259-1271 (2003) · Zbl 1063.34055
[17] Li, W. T.; Quan, H. S., Oscillation of higher order neutral differential equations with positive and negative coefficients, Ann. Differ. Equat., 11, 1, 70-76 (1995) · Zbl 0918.34068
[18] Li, Q. L.; Liang, H. Y.; Dong, W. L.; Zhang, Z. G., Existence of nonoscillatory solutions of higher-order difference equations with positive and negative coefficients, Bull. Korean Math. Soc., 45, 1, 23-31 (2008) · Zbl 1161.39006
[19] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G., Oscillation Theory of Differential Equations with Deviating Arguments (1987), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0832.34071
[20] Şahiner, Y.; Zafer, A., Bounded oscillation of nonlinear neutral differential equations of arbitrary order, Czechoslovak Math. J., 51, 126, 185-195 (2001) · Zbl 1079.34540
[21] Zhou, Y.; Zhang, B. G., Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients, Appl. Math. Lett., 15, 7, 867-874 (2002) · Zbl 1025.34065
[22] Zhou, Y.; Zhang, B. G., Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients, Comput. Math. Appl., 45, 6-9, 991-1000 (2003) · Zbl 1052.39015
[23] Zhu, Z. Q.; Wang, Q. R., Existence of nonoscillatory solutions to neutral dynamic equations on time scales, J. Math. Anal. Appl., 335, 1751-1762 (2007)
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