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On an asymptotic formula of Hartman in the theory of second order linear equations. (Russian) Zbl 0657.34056

The asymptotic formulae of P. Hartman [Trans. Am. Math. Soc. 63, 560-580 (1948; Zbl 0031.39801)], \(y_ 1\sim \exp (A(x))\), \(y_ 2\sim x \exp (-A(x))\), \(x\to \infty\), for the linearly independent solutions to the equation \(y''+a(x)y=0,\) \(a\in C[x_ 0,\infty)\), are valid if (1) \(\int^{\infty}_{x_ 0}x^{2q-1} | a(x)|^ q dx<\infty\) for some \(q\in [1,2]\). Here \(A(x)=\int^{x}_{x_ 0}ta(t)dt.\) The author shows that for every \(q<1\) or \(q>2\) there exists \(a\in C[x_ 0,\infty)\) satisfying (1) such there is no fundamental system \(y_ 1,y_ 2\) admitting the above asymptotic representations.
Reviewer: D.Bainov

MSC:

34E05 Asymptotic expansions of solutions to ordinary differential equations

Citations:

Zbl 0031.39801
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