OSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DYNAMIC EQUATIONS WITH MIXED NONLINEARITIES

TitleOSCILLATION CRITERIA FOR HIGHER ORDER NONLINEAR FUNCTIONAL DYNAMIC EQUATIONS WITH MIXED NONLINEARITIES
Publication TypeJournal Article
Year of Publication2015
AuthorsAGARWAL, RAVIP, GRACE, SAIDR, HASSAN, TAHERS
Volume19
Issue3
Start Page369
Pagination33
Date Published2015
ISSN1083-2564
AMS34K11, 39A10, 39A99
Abstract

In this paper, we will consider the higher-order functional dynamic equations with mixed nonlinearities of the form $${ \left\{  r_{n−1}(t) \phi_{αn−1} [ (r_{n−2}(t)(· · ·(r_1(t)\phi_{α1} [x^∆(t)])^∆ · · ·)^∆)^∆] \right.\} ^∆ + \sum^{N}_{ j=0} p_j (t)\phi_{γj} (x^σ (g_j (t))) = 0, }$$ on an above-unbounded time scale ${ \mathbb{T} }$, where ${ n ≥ 2 }$, and ${ \phi_β(u) := |u|^{β−1} \  u, β > 0. }$  The funtions ${ g_j : \mathbb{T} → \mathbb{T} }$ are rd-continuous functions such that ${ lim_{t→∞} g_j (t) = ∞, \  j = 0, 1, . . . , N. }$ The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

URLhttp://www.acadsol.eu/en/articles/19/3/5.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] R. P. Agarwal, M. Bohner and S. H. Saker, Oscillation of second order delay dynamic equation,
Canadian Appl. Math. Quart. 13 (2005), 1–17.
[2] R. P. Agarwal, D. O’Regan and S. H. Saker, Philos- type oscillation criteria for second order
half linear dynamic equations, Rocky Mountain J. Math. 37 (2007), 1085–1104.
[3] E. F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1961.
[4] M. Bohner and T. S. Hassan, Oscillation and boundedness of solutions to first and second order
forced functional dynamic equations with mixed nonlinearities, Appl. Anal. Discrete Math. 3 (2009), 242–252.
[5] M. Bohner, L Erbe and A. Peterson, Oscillation for nonlinear second order dynamic equations
on a time scale, J. Math. Anal. Appl. 301 (2005), 491–507.
[6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
[7] M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales,
Birkhauser, Boston, 2003.
[8] D. Chen, Oscillation and asymptotic behavior of solutions of certain third-order nonlinear
delay dynamic equations, Theoretical Mathematics & Applications 3 (2013), 19–33.
[9] E. M. Elabbasy and T. S. Hassan, Oscillation criteria for third order functional dynamic
equations, Electron. J. Differential Equations 2010 (2010), 1–14.
[10] L. Erbe, J. Baoguo and A. Peterson, Oscillation of nth-order superlinear dynamic equations
on time scales, Rocky Mountain J. Math. 41 (2011), 471–491.
[11] L. Erbe and T. S. Hassan, New oscillation criteria for second order sublinear dynamic equations,
Dynam. Syst. Appl. 22 (2013), 49–64.
[12] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for half-linear delay
dynamic equations on time scales, Nonlinear Dynam. Sys. Th. 9 (2009), 51–68.
[13] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for sublinear half-linear
delay dynamic equations on time scales, Int. J. Difference Equ. 3 (2008), 227–245.
[14] L. Erbe, A. Peterson and S. H. Saker, Hille-Kneser-type criteria for second-order dynamic
equations on time scales, Adv. Diff. Eq. 2006 (2006), 1–18.
[15] L. Erbe, A. Peterson and S. H. Saker, Hille and Nehari type criteria for third order dynamic
equations, J. Math. Anal. Appl. 329 (2007), 112–131.
[16] L. Erbe, A. Peterson and S. H. Saker, Asymptotic behavior of solutions of a third-order
nonlinear dynamic equation on time scales, J. Comp. Appl. Math. 181 (2005), 92–102.
[17] L. Erbe, A. Peterson and S. H. Saker, Oscillation and asymptotic behavior of a third-order
nonlinear dynamic equation, Canad. Appl. Math. Quarterly 14 (2006), 129–147.
[18] L. Erbe, T. S. Hassan and A. Peterson, Oscillation of third order nonlinear functional dynamic
equations on time scales, Differ. Equ. Dyn. Syst. 18 (2010), 199–227.
[19] L. Erbe, T. S. Hassan and A. Peterson, Oscillation criteria for nonlinear damped dynamic
equations on time scales, Appl. Math. Comput. 203 (2008), 343–357.
[20] L. Erbe, T. S. Hassan and A. Peterson, Oscillation of second order functional dynamic equations,
Int. J. Difference Equ. 5 (2010), 1–19.
[21] L. Erbe, T. S. Hassan and A. Peterson, Oscillation of third order functional dynamic equations
with mixed arguments on time scales, J. Appl. Math Comput. 34 (2010), 353–371.
[22] L. Erbe, B. Karapuz and A. Peterson, Kamenev-type oscillation criteria for higher order neutral
delay dynamic equations, Int. J. Differ. Equ. Appl. 6 (2011), 1–16.
[23] L. Erbe, R. Mert, A. Peterson and A. Zafer, Oscillation of even order nonlinear delay dynamic
equations on time scales, Czech. Math. J. 63 (2013), 265–279.
[24] M. Gera, J. R. Graef, M. Gregus, On oscillatory and asymptotic properties of solutions of
certain nonlinear third order differential equations, Nonlinear Anal. 32 (1998) 417–425, 49– 56.
[25] S. R. Grace, R. P. Agarwal, M. Bohner and D. O’Regan, Oscillation of second order strongly
superlinear and strongly sublinear dynamic equations, Commun. Nonlin. Sci. Numer. Simul.
14 (2009), 3463–3471.
[26] S. R. Grace, R. P. Agarwal and A. Zafer, Oscillation of higher order nonlinear dynamic equations
on time scale, Adv. Differ. Equ. 2012, 2012:67.
[27] S. R. Grace, J. Graef, S. Panigrahi and E. Tunc, On the oscillatory behavior of even order
neutral delay dynamic equations on time scales, Electron. J. Qual. Theory Differ. Equ. 2012
(2012), 1–12.
[28] S. R. Grace and T. S. Hassan, Oscillation criteria for higher order nonlinear dynamic equations,
Math. Nachr. (2014)/DOI 10.1002/mana.201300157, 1–15.
[29] S. R. Grace, On the oscillation of nth-order dynamic equations on time scale, Mediterr. J.
Math. 10 (2013), 147–156.
[30] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press,
Cambridge, 1988.
[31] Z. Han, T. Li, S. Sun and M. Zhang, Oscillation behavior of solutions of third-order nonlinear
delay dynamic equations on time scales, Commun. Korean Math. Soc. 26 (2011), 499–513.
[32] T. S. Hassan, Oscillation of third order nonlinear delay dynamic equations on time scales,
Math. Comput. Modelling 49 (2009), 1573–1586.
[33] T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math.
Anal. Appl. 345 (2008), 176–185.
[34] T. S. Hassan, Kamenev-type oscillation criteria for second order nonlinear dynamic equations
on time scales, Appl. Math. Comput. 217 (2011), 5285–5297.
[35] T. S. Hassan, L. Erbe, and A. Peterson, Forced oscillation of second order functional differential
equations with mixed nonlinearities, Acta Math. Sci. 31B (2011), 613–626.
[36] T. S. Hassan and Q. Kong, Oscillation criteria for second order nonlinear dynamic equations
with p-laplacian and damping, Acta Math. Sci. 33 (2013), 975–988.
[37] T. S. Hassan and Q. Kong, Interval criteria for forced oscillation of differential equations with
p-Laplacian, damping, and mixed nonlinearities, Dynam. Syst. Appl. 20 (2011), 279–294.
[38] S. Hilger, Analysis on measure chains — a unified approach to continuous and discrete calculus,
Results Math. 18 (1990), 18–56.
[39] V. Kac and P. Chueng, Quantum Calculus, Universitext, 2002.
[40] B. Karapuz, Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral
type with oscillating coefficients, Electron. J. Qual. Theory Differ. Equ. 34 (2009), 1–14.
[41] I. T. Kiguradze, On oscillatory solutions of some ordinary differential equations, Soviet Math.
Dokl. 144 (1962) 33–36.
[42] J. V. Manojlovic, Oscillation criteria for second-order half-linear differential equations, Math.
Comp. Mod. 30 (1999), 109–119.
[43] R. Mert, Oscillation of higher order neutral dynamic equations on time scales, Adv. Differ.
Equ. (2012) 2012:68.
[44] A. Ozbekler and A. Zafer, Oscillation of solutions of second order mixed nonlinear differential
equations under impulsive perturbations, Comput. Math. Appl. 61 (2011), no. 4, 933–940.
[45] S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales,
J. Comp. Appl. Math. 177 (2005), 375–387.
[46] Y. G. Sun and J. S. Wong, Oscillation criteria for second order forced ordinary differential
equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007), 549–560.
[47] T. Sun, W. Yu and H. Xi, Oscillatory behavior and comparison for higher order nonlinear
dynamic equations on time scales, J. Appl. Math. & Informatics 30 (2012), 289–304.
[48] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951), 368–380.
[49] Z. Yu and Q. Wang, Asymptotic behavior of solutions of third-order nonlinear dynamic equations,
J. Comp. Appl. Math. 225 (2009), 531–540.