Leighton-Wintner theorem

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Theorem (Leighton-Wintner theorem): Consider the self-adjoint equation $(py^{\Delta})^{\Delta}+qy^{\sigma}=0$. Assume $a \in \mathbb{T}, p>0, \sup \mathbb{T}=\infty,$ and $$\displaystyle\int_a^{\infty} \dfrac{1}{p(t)} \Delta t = \displaystyle\int_a^{\infty} q(t) \Delta t = \infty.$$ Then the self-adjoint equation is oscillatory on $[a,\infty)$.

Proof: