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Conjugacy and disconjugacy criteria for second order linear ordinary differential equations. (English) Zbl 1054.34053

The authors establish new conjugacy and disconjugacy criteria for the second-order linear differential equation \[ u''+p(t)u=0,\quad t\in \mathbb{R},\tag{*} \] where the function \(p\) is supposed to be locally integrable. Recall that (*) is said to be conjugate in an interval \(I\subseteq \mathbb{R}\) if there exists a nontrivial solution of this equation having at least two zeros in \(I\), and it is said to be disconjugate in the opposite case.
A typical result is the following statement. Let \(p(t)\not \equiv 0\) on \(\mathbb{R}\), \[ c(t):={1\over | t| }\int _0^t\int _0^s p(\xi )\,d\xi \,ds \] and suppose that \(c(-\infty ):=\lim _{t\to -\infty }c(t)\), \(c(\infty ):= \lim _{t\to \infty } c(t)\) exist and are finite. Equation (*) is conjugate on \(\mathbb{R}\) provided \(c(-\infty )+c(\infty ) \geq 0.\)
Some other (more complicated) conjugacy and disconjugacy criteria for (*) are given as well.

MSC:

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