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

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

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.**E. Akin, L. Erbe, B. Kaymakçalan, and A. Peterson, Oscillation results for a dynamic equation on a time scale, J. Differential Equations Appl. 7, (2001), 793-810. MR**2003d:39002****2.**M. Bohner, O. Doslý, and W. Kratz, An oscillation theorem for discrete eigenvalue problems, Rocky Mountain J. Math, (2002), to appear.**3.**M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. MR**2002c:34002****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.**O. Doslý and S. Hilger, A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Special Issue on ``Dynamic Equations on Time Scales'', edited by R. P. Agarwal, M. Bohner, and D. O'Regan, J. Comp. Appl. Math. 141(1-2), (2002) 147-158. MR**2003f:39015****6.**L. Erbe, Oscillation theorems for second order nonlinear differential equations, Proc. Amer. Math. Soc., 24 (1970), 811-814. MR**40:5973****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.**L. Erbe and A. Peterson, Riccati equations on a measure chain, In G. S. Ladde, N. G. Medhin, and M. Sambandham, editors,*Proceedings of Dynamic Systems and Applications*, volume 3, pages 193-199, Dynamic Publishers, Atlanta, GA, 2001. MR**2002h:34018****10.**L. Erbe and A. Peterson. Oscillation criteria for second-order matrix dynamic equations on a time scale, 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), 169-185. MR**2003e:34023****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.

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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:
https://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