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Existence of conjugate points for second-order linear differential equations. (English) Zbl 0820.34018

The paper deals with the equation (1) \(u''= p(t) u\), where \(-\infty< a< b< \infty\) and \(p: (a, b)\to \mathbb{R}\) is Lebesgue integrable on compact subsets of \((a, b)\). The function \(p(t)\) belongs to the class \(O((a,b))\) if (2) \(\int^ b_ a (s- a)(b- s)| p(s)| ds< \infty\) is satisfied and the solution \(u(t)\) of (1), satisfying the conditions (3) \(u(a+)= 0\), \(u'(a+)= 1\), has at least one zero on \((a, b)\). The function \(p(t)\) belongs to the class \(O'((a, b))\) if (4) \(\int^ b_ a (s- a)| p(s)| ds< \infty\) and the derivative \(u'(t)\) of the solution of problem (1), (3) has at least one zero on \((a, b]\). The sufficient conditions are given for \(p(t)\in O((a, b))\) (Theorem 1), for \(p(t)\not\in O((a, b))\) (Theorem 2) and for \(p(t)\in O'((a, b))\).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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References:

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