## 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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