Conjugacy criteria and principal solutions of selfadjoint differential equations. (English) Zbl 0841.34033

Authors’ abstract: “Oscillation properties of the selfadjoint two-term differential equation \[ (-1)^n (p(x) y^{(n)})^{(n)}+ q(x) y= 0\tag \(*\) \] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that \((*)\) is conjugate on \(\mathbb{R}= (-\infty, \infty)\) if there exist an integer \(m\in \{0, 1,\dots, n- 1\}\) and \(c_0, \dots, c_m\in \mathbb{R}\) such that \[ \int^0_\infty x^{2(n- m- 1)} p^{- 1}(x) dx= \infty= \int^\infty_0 x^{2(n- m- 1)} p^{- 1}(x)dx \] and \[ \limsup_{x_1\downarrow- \infty, x_2\uparrow\infty} \int^{x_2}_{x_1} q(x) (c_0+ c_1 x+\cdots+ c_m x^m)^2\leq 0, \quad q(x)\not\equiv 0. \] Some extensions of this criterion are suggested”.
Reviewer: F.Neuman (Brno)


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: EuDML