Boundedness and oscillation for nonlinear dynamic equations on a time scale

Authors:
Lynn Erbe and Allan Peterson

Journal:
Proc. Amer. Math. Soc. **132** (2004), 735-744

MSC (2000):
Primary 39A10

Published electronically:
July 14, 2003

MathSciNet review:
2019950

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation

on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient . We illustrate the results by several examples, including a nonlinear Emden-Fowler dynamic equation.

**1.**Elvan Akin, Lynn Erbe, Allan Peterson, and Billur Kaymakçalan,*Oscillation results for a dynamic equation on a time scale*, J. Differ. Equations Appl.**7**(2001), no. 6, 793–810. On the occasion of the 60th birthday of Calvin Ahlbrandt. MR**1870722**, 10.1080/10236190108808303**2.**M. Bohner, O. Doslý, and W. Kratz, An oscillation theorem for discrete eigenvalue problems, Rocky Mountain J. Math, (2002), to appear.**3.**Martin Bohner and Allan Peterson,*Dynamic equations on time scales*, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. MR**1843232****4.**M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain Journal of Mathematics, to appear.**5.**Ondřej Došlý and Stefan Hilger,*A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales*, J. Comput. Appl. Math.**141**(2002), no. 1-2, 147–158. Dynamic equations on time scales. MR**1908834**, 10.1016/S0377-0427(01)00442-3**6.**Lynn Erbe,*Oscillation theorems for second order nonlinear differential equations.*, Proc. Amer. Math. Soc.**24**(1970), 811–814. MR**0252756**, 10.1090/S0002-9939-1970-0252756-X**7.**L. Erbe, Oscillation criteria for second order linear equations on a time scale, Canadian Applied Mathematics Quarterly, 9 (2001), 1-31.**8.**L. Erbe, L. Kong and Q. Kong, Telescoping principle for oscillation for second order differential equations on a time scale, preprint.**9.**Lynn Erbe and Allan Peterson,*Riccati equations on a measure chain*, Dynamic systems and applications, Vol. 3 (Atlanta, GA, 1999) Dynamic, Atlanta, GA, 2001, pp. 193–199. MR**1864678****10.**Lynn Erbe and Allan Peterson,*Oscillation criteria for second-order matrix dynamic equations on a time scale*, J. Comput. Appl. Math.**141**(2002), no. 1-2, 169–185. Dynamic equations on time scales. MR**1908836**, 10.1016/S0377-0427(01)00444-7**11.**L. Erbe, A. Peterson, and P. Rehak, Comparison Theorems for Linear Dynamic Equations on Time Scales, Journal of Mathematical Analysis and Applications, 275 (2002), 418-438.**12.**L. Erbe, A. Peterson, and S. H. Saker, Oscillation Criteria for second-order nonlinear dynamic equations on time scales, Journal of the London Mathematical Society, 67 (2003), 701-714.**13.**S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universität Augsburg, 1999.**14.**C. Pötzsche, Chain rule and invariance principle on measure chains, Special Issue on ``Dynamic Equations on Time Scales'', edited by R. P. Agarwal, M. Bohner, and D. O'Regan, J. Comput. Appl. Math., 141(1-2) (2002), 249-254.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
39A10

Retrieve articles in all journals with MSC (2000): 39A10

Additional Information

**Lynn Erbe**

Affiliation:
Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323

Email:
lerbe@math.unl.edu

**Allan Peterson**

Affiliation:
Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323

Email:
apeterso@math.unl.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07061-8

Received by editor(s):
June 27, 2002

Received by editor(s) in revised form:
October 21, 2002

Published electronically:
July 14, 2003

Additional Notes:
This research was supported by NSF Grant 0072505

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2003
American Mathematical Society