Browse > Article
http://dx.doi.org/10.14317/jami.2018.213

OSCILLATION OF SECOND ORDER SUBLINEAR NEUTRAL DELAY DYNAMIC EQUATIONS VIA RICCATI TRANSFORMATION  

SETHI, ABHAY KUMAR (Department of Mathematics, Sambalpur University)
Publication Information
Journal of applied mathematics & informatics / v.36, no.3_4, 2018 , pp. 213-229 More about this Journal
Abstract
In this work, we establish oscillation of the second order sublinear neutral delay dynamic equations of the form:$$(r(t)((x(t)+p(t)x({\tau}(t)))^{\Delta})^{\gamma})^{\Delta}+q(t)x^{\gamma}({\alpha}(t))+v(t)x^{\gamma}({\eta}(t))=0$$ on a time scale T by means of Riccati transformation technique, under the assumptions $${\displaystyle\smashmargin{2}{\int\nolimits^{\infty}}_{t_0}}\({\frac{1}{r(t)}}\)^{\frac{1}{\gamma}}{\Delta}t={\infty}$$, and ${\displaystyle\smashmargin{2}{\int\nolimits^{\infty}}_{t_0}}\({\frac{1}{r(t)}}\)^{\frac{1}{\gamma}}{\Delta}t$ < ${\infty}$, for various ranges of p(t), where 0 < ${\gamma}{\leq}1$ is a quotient of odd positive integers.
Keywords
Oscillation; nonoscillation; neutral; delay dynamic equations; sublinear; time scale;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R.P. Agarwal, M. Bohner, D. O'Regan, A. Peterson, Dynamic equations on time scales: a survey, J. Math. Anal. Appl. Math. 141, N0.1-2 (2002), 1-26.
2 R.P. Agarwal, D. O'Regan, S.H. Saker, Oscillation criteria for second order nonlinear delay dynamic equations, J. Math. Anal. Appl. 300(2004), 203-217.   DOI
3 R.P. Agarwal, D. O'Regan, S.H. Saker, Oscillation criteria for nonlinear perturbed dynamic equations of second order on time scales, J. Appl. Math. Compu. 20(2006), 133-147.   DOI
4 M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with Applications, Birkhauser, Boston.,(2001).
5 M. Bohner, A. Peterson, Advance in dynamic equations on time scales, Birkhauser, Boston.,(2001).
6 M. Bohner, S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky. Mount. J. Math. 34(2004), 1239-1254.   DOI
7 L.H. Erbe, T.S. Hassan, A. Peterson, Oscillation criteria for sublinear half linear delay dynamic equations, Int. J. Difference Equ. 3(2008), 227-245.
8 S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results in mathematics 18, N0,1-2 .,(1990), 18-56.   DOI
9 S.H. Saker, Oscillation of second order nonlinear neutral delay dynamic equations on time scales, J. Comp. Appl. Math. 2(2006), 123-141.
10 S.H. Saker, Oscillation Theory of Dynamic Equations on Time Scales: Second and Third Orders, Lap. Lambert. Pub. Simulate. 24(2010),
11 S.H. Saker, D. O'Regan, New oscillation criteria for second order neutral functional dynamic equations via the general Riccati substitution, Commun. Sci. Numer. Simulate. 16(2011), 423-434.   DOI
12 A.K. Tripathy, Some oscillation results of second order nonlinear dynamic equations of neutral type, Nonlinear Analysis 71(2009), e1727-e1735.   DOI
13 A.K. Tripathy, G.B. Tenali, Oscillation results for second order neutral delay dynamic equations, J. Functional Diff. Equs. 17(2010), 329-344.
14 A.K. Tripathy, Riccati transformation and sublinear oscillation for second order neutral delay dynamic equations, J.Appl.Math and Informatics 30(2012), 1005-1021.
15 A.K. Sethi, Oscillation of a Functional Differential Equations of Second Order, Global. J. Pure and Appl. Math. 13(2017),5625-5634.
16 A.K. Sethi, A.K. Tripathy, Riccati transformation and oscillation of superlinear second order functional differential equations, Proceeding of Mathematics and Numerical aspects of Dynamical System., (2017),479-490.
17 T.S. Hassan, Oscillation criteria for half linear dynamic equations, J. Math. Anal.Appl. 345(2008), 176-185.   DOI
18 A.K. Tripathy, A.K. Sethi, Necessary and sufficient condition for oscillation of a class of second order neutral differential equations (communicated).
19 A.K. Tripathy, A.K. Sethi, Oscillation of sublinear and superlinear second order neutral differential equations, Int. J. Pure and Appl. Math. 113(2017), 73-71.