Boundedness and oscillation for nonlinear dynamic equations on a time scale

Erbe, Lynn; Peterson, Allan

doi:10.1090/S0002-9939-03-07061-8

Boundedness and oscillation for nonlinear dynamic equations on a time scale
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by Lynn Erbe and Allan Peterson PDF

Proc. Amer. Math. Soc. 132 (2004), 735-744 Request permission

Abstract:

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation \[ (p(t)x^{\Delta }(t))^{\Delta }+q(t)(f\circ x^{\sigma })=0, \] on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient $q(t)$. We illustrate the results by several examples, including a nonlinear Emden–Fowler dynamic equation.

References

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, DOI 10.1080/10236190108808303
M. Bohner, O. Došlý, and W. Kratz, An oscillation theorem for discrete eigenvalue problems, Rocky Mountain J. Math, (2002), to appear.
Martin Bohner and Allan Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. MR 1843232, DOI 10.1007/978-1-4612-0201-1
M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain Journal of Mathematics, to appear.
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, DOI 10.1016/S0377-0427(01)00442-3
Lynn Erbe, Oscillation theorems for second order nonlinear differential equations, Proc. Amer. Math. Soc. 24 (1970), 811–814. MR 252756, DOI 10.1090/S0002-9939-1970-0252756-X
L. Erbe, Oscillation criteria for second order linear equations on a time scale, Canadian Applied Mathematics Quarterly, 9 (2001), 1–31.
L. Erbe, L. Kong and Q. Kong, Telescoping principle for oscillation for second order differential equations on a time scale, preprint.
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
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, DOI 10.1016/S0377-0427(01)00444-7
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.
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.
S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Ph.D. thesis, Universität Augsburg, 1999.
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.