zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 for nonlinear second order dynamic equations on a time scale. (English) Zbl 1061.34018
Summary: We obtain some oscillation criteria for solutions to the nonlinear dynamic equation $$x^{\Delta\Delta}+ q(t)x^{\Delta^\sigma} +p(t) (f\circ x^\sigma)=0$$ on time scales. In particular, no explicit sign assumptions are made with respect to the coefficients $p(t)$, $q(t)$. We illustrate the results by several examples, including a superlinear Emden-Fowler dynamic equation.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
39A11Stability of difference equations (MSC2000)
Full Text: DOI
[1] Atkinson, F. V.: On second-order nonlinear oscillations. Pacific J. Math. 5, 643-647 (1955) · Zbl 0065.32001
[2] E. Akın-Bohner, M. Bohner, S.H. Saker, Oscillation criteria for a certain class of second order Emden -- Fowler dynamic equations, preprint · Zbl 1177.34047
[3] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications. (2001) · Zbl 0978.39001
[4] M. Bohner, S.H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math., in press · Zbl 1075.34028
[5] M. Bohner, S.H. Saker, Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comput. Modelling, in press · Zbl 1112.34019
[6] Erbe, L.: Oscillation theorems for second order nonlinear differential equations. Proc. amer. Math. soc. 24, 811-814 (1970) · Zbl 0194.12102
[7] Erbe, L.: Oscillation criteria for second order linear equations on a time scale. Canad. appl. Math. quart. 9, 1-31 (2001) · Zbl 1050.39024
[8] Erbe, L.: Oscillation criteria for second order nonlinear differential equations. Ann. mat. Pura appl. 44, 257-268 (1972) · Zbl 0296.34026
[9] Erbe, L.; Peterson, A.: Oscillation criteria for second order matrix dynamic equations on a time scale. J. comput. Appl. math. 141, 169-185 (2002) · Zbl 1017.34030
[10] Erbe, L.; Peterson, A.: Boundedness and oscillation for nonlinear dynamic equations on a time scale. Proc. amer. Math. soc. 132, 735-744 (2003) · Zbl 1055.39007
[11] L. Erbe, A. Peterson, An oscillation result for a nonlinear dynamic equation on a time scale, Canad. Appl. Math. Quart., in press · Zbl 1086.39004
[12] Erbe, L.; Peterson, A.; Rehak, P.: Comparison theorems for linear dynamic equations on time scales. J. math. Anal. appl. 275, 418-438 (2002) · Zbl 1034.34042
[13] Erbe, L.; Peterson, A.; Saker, S. H.: Oscillation criteria for second-order nonlinear dynamic equations on time scales. J. London math. Soc. 67, 701-714 (2003) · Zbl 1050.34042
[14] Fite, W. B.: Concerning the zeros of solutions of certain differential equations. Trans. amer. Math. soc. 19, 341-352 (1917) · Zbl 46.0702.01
[15] S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universität Augsburg, 1999
[16] Leighton, W.: On self-adjoint differential equations of second order. J. London math. Soc. 27, 37-47 (1952) · Zbl 0048.06503
[17] Pötzsche, C.: Chain rule and invariance principle on measure chains. J. comput. Appl. math. 141, 249-254 (2002) · Zbl 1011.34045
[18] Waltman, P.: An oscillation criterion for a nonlinear second order equation. J. math. Anal. appl. 10, 439-441 (1965) · Zbl 0131.08902
[19] Wintner, A.: On the nonexistence of conjugate points. Amer. J. Math. 73, 368-380 (1951) · Zbl 0043.08703