Asymptotic equivalence of differential equations and asymptotically almost periodic solutions |
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Authors: | MU Akhmet MA Tleubergenova A Zafer |
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Institution: | 1. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara, Turkey;2. Department of Mathematics, Aktobe State Pedagogical University, 463000 Aktobe, pr. Moldagulovoy, 34, Kazakhstan;3. Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey |
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Abstract: | In this paper we use Rab’s lemma M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires, 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, V.V. 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. |
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Keywords: | 34A30 34C27 34C41 34D10 |
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