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Keywords:
conjugate points; principal system of solutions; variational method; conjugacy criteria
conjugate points; principal system of solutions; variational method; conjugacy criteria
Summary:
Oscillation properties of the self-adjoint, two term, differential equation \[(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=0\qquad \mathrm {(*)}\] are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \lbrace 0,1,\dots ,n-1\rbrace $ and $c_0,\dots ,c_m\in R$ such that \[\int _\infty ^0 x^{2(n-m-1)}p^{-1}(x)\,dx=\infty =\int _0^\infty x^{2(n-m-1)}p^{-1}(x)\,dx\] and \[\limsup _{x_1\downarrow -\infty ,x_2\uparrow \infty }\int _{x_1}^{x_2}q(x)(c_0+c_1x+\dots + c_mx^m)^2\,dx\le 0,\quad q(x)\lnot \equiv 0.\] Some extensions of this criterion are suggested.
References:
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