Weighted differential inequality and oscillatory properties of fourth order differential equations

Abstract

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.

On oscillatory and spectral properties of a class of higher-order differential operators

“…which yields the left estimate in (8). From (31) we get the left estimate in (9). The proof of Theorem 1 is complete.…”

“…Repeating the proof of the necessary part of Theorem 1 in [9], for each g 1 ∈ L 1 we construct a function g 2 ∈ L 2 and for each g 2 ∈ L 2 we construct a function g 1 ∈ L 1 so that g(t) = g 1 (t) for 0 < t ≤ τ and g(t) = g 2 (t) for t > τ belongs to the set L p,v (I). For the constructed function g we have (see [9, (26)])…”

“…Similarly, we find that lim τ →0 + F 2 2 (τ ) = 0, i.e., lim τ →0 + F 2 (τ ) = 0. From the right estimate in (9) we have…”

Oscillatory and spectral properties of fourth-order differential operator and weighted differential inequality with boundary conditions

“…This variational method allows to remove restrictions on the coefficients. Using the variational method, the paper [23] studies oscillation of equations in form (8) in the standard case, while the papers [15] and [16] discuss the problem in the overdetermined cases. The spectral properties of the operator of the fourth and higher orders generated by the differential expression in form (9) are considered, e.g., in the works [3,4], [9, Chaps.…”