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In the paper, we establish the oscillatory and spectral properties of a class of fourth-order differential operators in dependence on integral behavior of its coefficients at zero and infinity. In order to obtain these results, we investigate a certain weighted second-order differential inequality of independent interest.

In this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality.

Abstract. In the work we propose a new approach for studying the asymptotic behavior for large of the solutions to singular linear two-terms differential equationswith a potential ( ) growing non-regularly as → ∞. The idea of constructing the asymptotics for the solutions of singular linear differential equations and its effectiveness is demonstrated for 4 th order equations with an oscillating potential.

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