Sharp oscillation theorem for fourth-order linear delay differential equations
Irena Jadlovská, Jozef Džurina, John R. Graef, Said R. Grace
Abstract
Abstract In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation $$ x^{(4)}(t) + p(t)x\bigl(\tau (t)\bigr) = 0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mi>x</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.