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Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

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In this paper a system of differential equations y′ − A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A1+ A 0 +λ −1A−1(·,λ) and A 1,A 0, A − 1 are n × n matrix-functions. The main assumptions: A 1 is absolutely continuous on the interval [a, b], A 0 and A - 1(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ1(x),…,φn (x) of the characteristic equation det (φ E — A 1) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ⊂ C on which the inequalities Re(λφ1(x)) ≤ … ≤ Re (λφn(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φj(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A 1 (x), A o (x) and A - 1 (x).

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Authors and Affiliations

  1. Department of Mechanics & Mathematics, Saratov State University, Astrakhan-skaya str. 83, 410026, Saratov, Russia

    Viktor S. Rykhlov

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  1. Viktor S. Rykhlov
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Correspondence to Viktor S. Rykhlov.

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Research in part is supported by RFBR (Russia) under Grant no. 97-01-00566 and by Volkswagen Foundation (Germany) under Project “Inverse Probleme der Spektraltheorie und ihre Anwendungen” (1997).

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Rykhlov, V.S. Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order. Results. Math. 36, 342–353 (1999). https://doi.org/10.1007/BF03322121

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