Second-Order Differential Equation: Oscillation Theorems and Applications

Abstract

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ${\left(\varsigma \left(y\right){\left(u^{\prime }\left(y\right)\right)}^a\right)}^{\prime }+p\left(y\right)u^c\left(\vartheta \left(y\right)\right)=0,\text{for}y\ge y_0,$ under the assumption ${{\displaystyle \int }}^{\infty }{\left(\varsigma \left(\eta \right)\right)}^{-\left(1/a\right)}=\infty$ . Two cases are considered for $a<c$ and $a>c$ , where $a$ and $c$ are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.