Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter May 8, 2014

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

John Graef EMAIL logo , Saroj Panigrahi and P. Reddy
From the journal Mathematica Slovaca

Abstract

In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.

[1] BOHNER, M.— PETERSON, A.: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0201-110.1007/978-1-4612-0201-1Search in Google Scholar

[2] BOHNER, M.— PETERSON, A.: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. http://dx.doi.org/10.1007/978-0-8176-8230-910.1007/978-0-8176-8230-9Search in Google Scholar

[3] GRAEF, J. R.— GRAMMATIKOPOULOS, M. K.— SPIKES, P. W.: Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order, Nonlinear Anal. 21 (1993), 23–42. http://dx.doi.org/10.1016/0362-546X(93)90175-R10.1016/0362-546X(93)90175-RSearch in Google Scholar

[4] HILGER, S.: Analysis on measure chains: a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56. http://dx.doi.org/10.1007/BF0332315310.1007/BF03323153Search in Google Scholar

[5] KARPUZ, B.— ÖCALAN, Ö.: Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations, Nonlinear Anal. 71 (2009), 3063–3071. http://dx.doi.org/10.1016/j.na.2009.01.21810.1016/j.na.2009.01.218Search in Google Scholar

[6] PANIGRAHI, S.— REDDY, P. R.: On oscillatory fourth order nonlinear neutral delay dynamic equations, Comput. Math. Appl. 62 (2011), 4258–4271. http://dx.doi.org/10.1016/j.camwa.2011.10.01310.1016/j.camwa.2011.10.013Search in Google Scholar

[7] PARHI, N.— TRIPATHY, A. K.: On oscillatory fourth order nonlinear neutral differential equations II, Math. Slovaca 55 (2005), 183–202. Search in Google Scholar

[8] THANDAPANI, E.— AROCKIASAMY, I. M.: Oscillatory and asymptotic behaviour of fourth order non-linear delay difference equations, Indian J. Pure Appl. Math. 32 (2001), 109–123. Search in Google Scholar

[9] THANDAPANI, E.— SUNDARAM, P.— GRAEF, J. R.— SPIKES, P. W.: Asymptotic behavior and oscillation of solutions of neutral delay difference equations of arbitrary order, Math. Slovaca 47 (1997), 539–551. Search in Google Scholar

Published Online: 2014-5-8
Published in Print: 2014-4-1

© 2014 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 10.12.2022 from https://www.degruyter.com/document/doi/10.2478/s12175-014-0209-7/html
Scroll Up Arrow