Asymptotic equivalence of differential equations and asymptotically almost periodic solutions


Creative Commons License

Akhmet M. U. , Tleubergenova M. A. , Zafer A.

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, vol.67, no.6, pp.1870-1877, 2007 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 67 Issue: 6
  • Publication Date: 2007
  • Doi Number: 10.1016/j.na.2006.07.045
  • Title of Journal : NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
  • Page Numbers: pp.1870-1877

Abstract

In this paper we use Rab's lemma [M. Rab, Uber lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222-229; M. Rab, Note sur les formules asymptotiques pour les solutions d'un systeme d'equations differentielles lineaires, Czechoslovak Math. J. 91 (1966) 127-129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich's result [V.V. Nemytskii, VX Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217-240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed. (c) 2006 Elsevier Ltd. All rights reserved.