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Asymptotic behavior of increasing solutions to a system of n nonlinear differential equations

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    0385126 - MÚ 2013 RIV GB eng J - Journal Article
    Řehák, Pavel
    Asymptotic behavior of increasing solutions to a system of n nonlinear differential equations.
    Nonlinear Analysis: Theory, Methods & Applications. Roč. 77, January 12 (2013), s. 45-58. ISSN 0362-546X. E-ISSN 1873-5215
    Institutional support: RVO:67985840
    Keywords : oncreasing solution * asymptotic formula * quasilinear system
    Subject RIV: BA - General Mathematics
    Impact factor: 1.612, year: 2013

    We consider the system x(i)' = a(i)(t)vertical bar x(i+1)vertical bar(alpha i)sgn x(i+1), i = 1, ... , n, n = 2, where ai, i = 1,..., n, are positive continuous functions on [a, infinity), alpha(i) is an element of (0, infinity), i = 1,..., n, with alpha(1) ... alpha(n) < 1, and x(n+1) means x(1). We analyze the asymptotic behavior of the solutions to this system whose components are eventually positive increasing. In particular, we derive exact asymptotic formulas for the extreme case, where all the solution components tend to infinity (the so-called strongly increasing solutions). We offer two approaches: one is based on the asymptotic equivalence theorem, and the other utilizes the theory of regular variation. The above-mentioned system includes, as special cases, two-term nonlinear scalar differential equations of arbitrary order n and systems of n/2 second-order nonlinear equations (provided n is even), which are related to elliptic partial differential systems.
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