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

Abstract

Abstract In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality’s weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero.

Keywords

• Fourth-order differential equation
• Oscillation
• Nonoscillation
• Variational principle
• Weighted inequality