HILLE AND NEHARI TYPE CRITERIA FOR HIGHER ORDER FUNCTIONAL DYNAMIC EQUATIONS

TitleHILLE AND NEHARI TYPE CRITERIA FOR HIGHER ORDER FUNCTIONAL DYNAMIC EQUATIONS
Publication TypeJournal Article
Year of Publication2015
AuthorsERBE, LYNN, HASSAN, TAHERS
Volume19
Issue3
Start Page463
Pagination24
Date Published2015
ISSN1083-2564
AMS34K11, 39A10, 39A9
Abstract

In this paper, we study the higher order functional dynamic equation $$\left\{  r_{n−1}(t) ( r_{n−2}(t)\left(...(r_1(t)x^∆(t))^∆ ...)^∆ \right)^∆ \right\}^∆ + p (t) x (g (t)) = 0,$$ on a time scale ${ \mathbb{T} }$, which is unbounded above, and where ${ n ≥ 2}$ . We will extend the so-called Hille and Nehari type criteria to higher order dynamic equations on time scales. Our results are essentially new even for higher order differential and difference equations. Therefore, the results obtained extend and improve several known results in the literature on second-order and third-order dynamic equations. We illustrate these new results by means of several examples.

 

URLhttp://www.acadsol.eu/en/articles/19/3/11.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, M. Bohner, T. Li and C. Zhang, Hille and Nehari type criteria for third order
delay dynamic equations, J. Difference Equ. Appl. 19 2013. no. 10, 1563–1579.
[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), no. 2, 242–252.
[5] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkh¨auser, Boston, 2001
[6] M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
[7] D. Chen, Oscillation and asymptotic behavior of solutions of certain third-order nonlinear delay
dynamic equations, Theoretical Mathematics & Applications 3 (2013), 19–33.
[8] E. M. Elabbasy and T. S. Hassan, Oscillation criteria for third order functional dynamic equations,
Electron. J. Diff. Equ. 2010 (2010), no. 131 1–14.
[9] L. Erbe, Oscillation criteria for second order nonlinear delay equations, Canad. Math. Bull. 16 (1973), 49–56.
[10] L. Erbe, J. Baoguo and A. Peterson, Oscillation of nth-order superlinear dynamic equations on
time scales, Rocky Mountain J. Math. 41 (2011), no. 2, 471–491.
[11] L. Erbe and T. S. Hassan, New oscillation criteria for second order sublinear dynamic equations,
Dynam. Systems Appl. 22 (2013), no. 1, 49–64.
[12] 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), no. 1-2, 199–227.
[13] L. Erbe, T. S. Hassan and A. Peterson, Oscillation criteria for nonlinear damped dynamic
equations on time scales, Appl. Math. Comput. 203, (2008), no. 1, 343–357.
[14] L. Erbe, T. S. Hassan and A. Peterson, Oscillation criteria for nonlinear functional neutral
dynamic equations on time scales, J. Differerence Equ. Appl. 15 (2009), no. 11-12, 1097–1116.
[15] L. Erbe, T S. Hassan and A. Peterson, Oscillation of second order functional dynamic equations,
Int. J. Differerence Equ. 5, (2010), no. 2, 175–193.
[16] L. Erbe, T. S. Hassan, A. Peterson and S. H. Saker, Oscillation criteria for half-linear delay
dynamic equations on time scales, Nonlinear Dyn. Sys. Theory 9 (2009), no. 1, 51–68.
[17] 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), no. 2, 227-245.
[18] L. Erbe, A. Peterson and S. H. Saker, Asymptotic behavior of solutions of a third-order nonlinear
dynamic equation on time scales, J. Comput. Appl. Math. 181 (2005), no. 1, 92–102.
[19] L. Erbe, A. Peterson and S. H. Saker, Oscillation criteria for second-order nonlinear delay
dynamic equations on time scales, J. Math. Anal. Appl. 333 (2007), no. 1, 505–522.
[20] L. Erbe, A. Peterson and S. H. Saker, Hille and Nehari type criteria for third order dynamic
equations, J. Math. Anal. Appl. 329 (2007), no. 1, 112–131.
[21] L. Erbe, B. Karapuz and A. Peterson, Kamenev-type oscillation criteria for higher order neutral
delay dynamic equations, Int. J. Differnce Equ. 6 (2011), no. 1, 1–16.
[22] L. Erbe, R. Mert, A. Peterson and A. Zafer, Oscillation of even order nonlinear delay dynamic
equations on time scales, Czechoslovak Math. J. 63(138) (2013), no. 1, 265–279.
[23] W. B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer.
Math. Soc. 19 (1918), no. 4, 341–352.
[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), no. 3, 417–425.
[25] S. R. Grace and T. S. Hassan, Oscillation criteria for higher order nonlinear dynamic equations,
Math. Nachr. to appear.
[26] S. R. Grace, R. P. Agarwal, M. Bohner and D. O’Regan, Oscillation of second order strongly
superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul.
14 (2009), no. 8, 3463–3471.
[27] S. R. Grace, R. P. Agarwal and A. Zafer, Oscillation of higher order nonlinear dynamic equations
on time scale, Adv. Difference Equ. 2012, 2012:67, 18 pp.
[28] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press,
Cambridge, 1988.
[29] 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), no. 3, 499–513.
[30] T. S. Hassan, Oscillation of third order nonlinear delay dynamic equations on time scales, Math.
Comput. Modelling 49 (2009), no. 7-8, 1573–1586.
[31] T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math.
Anal. Appl. 345 (2008), no. 1, 176–185.
[32] T. S. Hassan, Kamenev-type oscillation criteria for second order nonlinear dynamic equations
on time scales, Appl. Math. Comput. 217 (2011), no. 12, 5285–5297.
[33] T. S. Hassan, Oscillation criteria for second order nonlinear dynamic equations, Adv. Difference
Equ. 2012 2012:171, 13 pp.
[34] T. S. Hassan, L. Erbe and A. Peterson, Oscillation of second order superlinear dynamic equations
with damping on time scales, Comput. Math. Appl. 59 (2010), no. 1, 550–558.
[35] T. S. Hassan and Q. Kong, Oscillation criteria for second order nonlinear dynamic equations
with p-Laplacian and damping, Acta Math. Sci. Ser. B Engl. Ed. 33 (2013), no. 4, 975–988.
[36] S. Hilger, Analysis on measure chains — a unified approach to continuous and discrete calculus,
Results Math. 18 (1990), no. 1-2, 18–56.
[37] E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252.
[38] V. Kac and P. Chueng, Quantum Calculus, Universitext, 2002.
[39] B. Karapuz, Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral
type with oscillating coefficients, Electron J. Qual. Theory Diff. Equ. 2009 (2009), no. 34, 1–14.
[40] I. T. Kiguradze, On oscillatory solutions of some ordinary differential equations, Soviet Math.
Dokl. 144 (1962), 33–36.
[41] W. Leighton, The detection of the oscillation of solutions of a second order linear differential
equation, Duke Math. J., 17 (1950), 57–62.
[42] Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math.
Soc. 85 (1957), 428–445.
[43] J. Ohriska, Oscillation of second order delay and ordinary differential equations, Czechoslovak
Math. J. 34 (1984), no. 1, 107–112.
[44] S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J.
Comput. Appl. Math. 177 (2005), no. 2, 375–387.
[45] T. Sun, W. Yu and H. Xi, Oscillatory behavior and comparison for higher order nonlinear
dynamic equations on time scales, J. Appl. Math. Inform. 30 (2012), no. 1-2, 289–304.
[46] Y. Wang and Z. Xu, Asymptotic properties of solutions of certain third order dynamic equations,
J. Comput. Appl. Math. 236 (2012), no. 9, 2354–2366.
[47] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951), 368–380.
[48] J. S. W. Wong, Second order oscillation with retarded arguments, in Ordinary differential
equations, 581–596, Washington, 1971, Academic press, New York and London, 1972.
[49] Z. Yu and Q. Wang, Asymptotic behavior of solutions of third-order nonlinear dynamic equations,
J. Comp. Appl. Math. 225 (2009), no. 2, 531–540.