115
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium

&
Pages 841-857 | Received 12 Jun 2009, Accepted 29 Dec 2009, Published online: 09 May 2011

References

  • Akhmetov , M. , Tleubergenova , M.A. and Zafer , A. 2007 . Asymptotic equivalence of differential equations and asymptotically almost periodic solutions . Nonlinear Anal. , 67 : 1870 – 1877 .
  • Atkinson , F.V. 1958 . On stability and asymptotic equilibrium . Ann. Math. , 68 : 690 – 708 .
  • Aulbach , B. and Neidhart , L. 2004 . “ Integration on measure chains ” . In Proceedings of the Sixth International Conference on Difference Equations , 239 – 252 . Boca Raton, FL : CRC .
  • Bellman , R. 1947 . On the boundedness of solutions of non-linear differential and difference equations . Trans. Am. Math. Soc. , 62 : 357 – 386 .
  • Bodine , S. and Lutz , D.A. 2007 . On asymptotic equivalence of perturbed linear systems of differential and difference equations . J. Math. Anal. Appl. , 326 : 1174 – 1189 .
  • Bohner , M. and Lutz , D.A. 2001 . Asymptotic behavior of dynamic equations on time scales . J. Differ. Equ. Appl. , 7 : 21 – 50 .
  • Bohner , M. and Peterson , A. 2001 . Dynamic Equations on Time Scales , Boston, MA : Birkhäuser .
  • Brauer , F. 1961 . Global behavior of solutions of ordinary differential equations . J. Math. Anal. Appl. , 2 : 145 – 158 .
  • Brauer , F. 1962 . Asymptotic equivalence and asymptotic behaviour of linear systems . Michigan Math. J. , 9 : 33 – 43 .
  • Brauer , F. and Wong , J.S.W. 1969 . On asymptotic behavior of perturbed linear systems . J. Differ. Equ. , 6 : 142 – 153 .
  • Choi , S.K. , Koo , N.J. and Ryu , H.S. 2003 . Asymptotic equivalence between two difference systems . Comput. Math. Appl. , 45 : 1327 – 1337 .
  • Hallam , T.G. and Heidel , J.W. 1970 . The asymptotic manifolds of a perturbed linear system of differential equations . Trans. Am. Math. Soc. , 149 : 233 – 241 .
  • Hartman , P. and Wintner , A. 1955 . Asymptotic integrations of ordinary non-linear differential equations . Am. J. Math. , 77 : 692 – 724 .
  • Hilger , S. 1990 . Analysis on measure chains – a unified approach to continuous and discrete calculus . Results Math. , 18 : 18 – 56 .
  • Kaymakçalan , B. , Lakshmikantham , V. and Sivasundaram , S. 1996 . Dynamic Systems on Measure Chains , Dordrecht : Kluwer .
  • Kaymakçalan , B. , Mert , R. and Zafer , A. 2007 . Asymptotic equivalence of dynamic systems on time scales . Discrete Contin. Dyn. Syst. , : 558 – 567 . Supplement Volume 2007
  • Ladas , G. , Lakshmikantham , V. and Leela , S. 1971 . On the perturbability of the asymptotic manifold of a perturbed system of differential equations . Proc. Am. Math. Soc. , 27 : 65 – 71 .
  • Medina , R. 1996 . Asymptotic properties of solutions of nonlinear difference equations . J. Comput. Appl. Math. , 70 : 57 – 66 .
  • Medina , R. 1998 . Asymptotic behavior of nonlinear difference systems . J. Math. Anal. Appl. , 219 : 294 – 311 .
  • Medina , R. and Pinto , M. 1997 . Asymptotic equivalence and asymptotic behavior of difference systems . Commun. Appl. Anal. , 1 : 511 – 523 .
  • Perron , O. 1929 . Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen . Math. Z. , 29 : 129 – 160 .
  • Rehak , P. 2004 . On certain comparison theorems for half-linear dynamic equations on time scales . Abstr. Appl. Anal. , 7 : 551 – 565 .
  • Saito , S. 1992 . Asymptotic equivalence of quasilinear ordinary differential systems . Math. Jpn. , 37 : 503 – 513 .
  • Trench , W.F. 1986 . Extensions of a theorem of Wintner on systems with asymptotically constant solutions . Trans. Am. Math. Soc. , 293 : 477 – 483 .
  • Trench , W.F. 1998 . Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems . Comput. Math. Appl. , 36 : 261 – 267 .
  • Trench , W.F. 2000 . Systems of difference equations with asymptotically constant solutions . Nonlinear Anal. , 40 : 611 – 615 .
  • Wintner , A. 1954 . On a theorem of Bôcher in the theory of ordinary linear differential equations . Am. J. Math. , 76 : 183 – 190 .
  • Zafer , A. 2005 . On asymptotic equivalence of linear and quasilinear difference equations . Appl. Anal. , 84 : 899 – 908 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.