New oscillation criteria for forced superlinear neutral type differential equations

Ethiraju Thandapani; Sivaraj Tamilvanan

doi:10.26637/mjm0101/009

Abstract

Some new oscillation criteria are established for the neutral type differential equation
$$
\left(a(t)\left((x(t)+p(t) x(\tau(t)))^{\prime}\right)^\alpha\right)^{\prime}+q(t) x^\beta(t)=e(t), t \geq t_0,
$$
which are applicable to equations with nonnegative forcing term. Examples are provided to illustrate the results.

Keywords:

Neutral differential equation, second order, oscillation, superlinear

Mathematics Subject Classification:

34C15

Authors Imprint References Funding Related Articles Downloads

Ethiraju Thandapani Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai-600005, India.
Sivaraj Tamilvanan Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai-600005, India.

Pages: 67-72
Date Published: 01-09-2012
Vol. 1 No. 1 (2012): Inaugural Issue :: Malaya Journal of Matematik (MJM)

R.P. Agarwal, S.R. Grace and D.O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht, 2000. DOI: https://doi.org/10.1007/978-94-015-9401-1

L.H. Erbe, A. Peterson, T.S. Hasan and S.H. Saker, Interval oscillation criteria for forced second order nonlinear delay dynamic equations with oscillatory potential, Dyn. Cont. Disc. Syst. A, 17(2010), 533-542.

L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory For Functional Differential Equations, Marcel Dekker, New York, 1995.

L.H. Erbe, A. Peterson and S.H. Saker, Oscillation criteria for a forced second order nonlinear dynamic equations, J. Diff. Eqns. Appl., 14(2008), 997-1009. DOI: https://doi.org/10.1080/10236190802332175

A.G. Kartsatos, On the maintenance of oscillations of n th order equations under the effect of a small forcing term, J. Differ. Equ., 10(1971), 355-363. DOI: https://doi.org/10.1016/0022-0396(71)90058-1

A.G. Kartsatos, Maintenance of oscillations under the effect of a periodic forcing term, Proc. Amer. Math. Soc., 33(1972), 377-383. DOI: https://doi.org/10.1090/S0002-9939-1972-0330622-0

Q. Kong and J.S.W. Wong, Oscillation of a forced second order differential equations with a deviating argument, Funct. Diff. Equ., 17(2010), 141-155.

Q. Kong and B.G. Zhang, Oscillation of a forced second order nonlinear equation, Chin. Ann. Math., 15B:1(1994), 59-68.

G.S. Ladde,V. Lakshmikantham and B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.

A.H. Nazar, Sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc., 126(1998), 123-125. DOI: https://doi.org/10.1090/S0002-9939-98-04354-8

Y.G. Sun and J.S.W. Wong, Forced oscillation of second order superlinear differential equations, Math. Nachr., 278(2005), 1621 -1628. DOI: https://doi.org/10.1002/mana.200410327

Y.G. Sun and J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl., 334(2007), 549-560. DOI: https://doi.org/10.1016/j.jmaa.2006.07.109

J.S.W.Wong, Second order nonlinear forced oscillations, SIAM J. Math. Anal., 19(1998), 667-675. DOI: https://doi.org/10.1137/0519047

Q.G. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equation with oscillatory potential, Appl. Math., 136(2003), 49-64. DOI: https://doi.org/10.1016/S0096-3003(01)00307-1