zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. (English) Zbl 1153.34040

Summary: By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

[r(t)[y(t)+p(t)y(τ(t))] Δ ] Δ +q(t)f(y(δ(t)))=0,

on a time scale 𝕋. The results improve some oscillation results for neutral delay dynamic equations and in the special case when 𝕋= our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math. 48, No. 4, 871–886 (1996; Zbl 0859.34055)]. When 𝕋=, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36, No. 10–12, 123–132 (1998; Zbl 0933.39027)]. When 𝕋=h,𝕋={t:t=qk,k,q>1}, 𝕋= 2 ={t 2 :t}, 𝕋=𝕋 n ={t n =Σ k=1 n 1 k,n 0 }, 𝕋={t 2 :t}, 𝕋={n:n 0 } and 𝕋={n 3:n 0 } our results are essentially new. Some examples illustrating our main results are given.

MSC:
34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
39A10Additive difference equations
References:
[1]Hilger, S.: Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math., 18, 18–56 (1990)
[2]Agarwal, R. P., Bohner, M. O’Regan, D., Peterson, A.: Dynamic equations on time scales: A survey. J. Comp. Appl. Math., Special Issue on Dynamic Equations on Time Scales, edited by R. P. Agarwal, M. Bohner, and D. O’Regan, (Preprint in Ulmer Seminare 5), 141(1–2), 1–26 (2002)
[3]Kac, V., Cheung, P.: Quantum Calculus, Springer, New York, 2001
[4]Bohner, M., Peterson, A.: Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, 2001
[5]Spedding, V.: Taming Nature’s Numbers. New Scientist. 19, 28–31 (2003)
[6]Agarwal, R. P., O’Regan, D., Saker, S. H.: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. and Appl., 300, 203–217 (2004) · Zbl 1062.34068 · doi:10.1016/j.jmaa.2004.06.041
[7]Saker, S. H.: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comp. Appl. Math., 187, 123–141 (2006) · Zbl 1097.39003 · doi:10.1016/j.cam.2005.03.039
[8]Şahiner, Y.: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Adv. Difference Eqns., 2006, 1–9 (2006)
[9]Wu, H., Wu, Zhuang, R. K., Mathsen, R. M.: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comp., 178, 321–331 (2006)
[10]Agarwal, R. P., O’Regan, D., Saker, S. H.: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Appl. Analysis, 86, 1–17 (2007) · Zbl 1151.01322 · doi:10.1016/j.jmaa.2007.03.059
[11]Saker, S. H.: Hille and Nehari types oscillation criteria for second-order neutral delay dynamic equations. Dyn. Cont. Disc. Imp. Sys, (accepted)
[12]Saker, S. H.: Oscillation of second-order delay and neutral delay dynamic equations on time scales. Dyn. Syst. & Appl., 16, 345–360 (2007)
[13]Mathsen, R. M., Wang, Q. R., Wu, H. W.: Oscillation for neutral dynamic functional equations on time scales. J. Diff. Eqns. Appl., 10, 651–659 (2004) · Zbl 1060.34038 · doi:10.1080/10236190410001667968
[14]Saker, S. H.: Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type. Dyn. Syst. & Appl., 15, 629–644 (2006)
[15]Li, H. J.: Oscillation criteria for second order linear differential equations. J. Math. Anal. Appl., 194, 312–321 (1995) · Zbl 0829.34060 · doi:10.1006/jmaa.1995.1173
[16]Li, H. J., Liu, W. L.: Oscillation criteria for second order neutral differential equations. Canad. J. Math., 48, 871–886 (1996) · Zbl 0859.34055 · doi:10.4153/CJM-1996-044-6
[17]Li, H. J., Yeh, C. C.: Oscillation criteria for second-order neutral delay difference equations. Comp. Math. Appl., 36, 123–132 (1998) · Zbl 0933.39027 · doi:10.1016/S0898-1221(98)80015-1
[18]Bohner, E. Akin, Bohner, M., Akin, F.: Pachpatte inequalities on time scales. JIPAM. J. Ineq. Pure Appl. Math., 6, 1–23 (2005)
[19]Bohner, M., Stević, S.: Asymptotic behavior of second-order dynamic equations. Appl. Math. Comp., 188, 1503–1512 (2007) · Zbl 1124.39003 · doi:10.1016/j.amc.2006.11.016