Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications

Abstract

Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the form ${\left(\varsigma \left(y\right){\left(u^{\prime }\left(y\right)\right)}^a\right)}^{\prime }+{\displaystyle {\sum }_{j=1}^m{p}_j\left(y\right)u^{c_j}\left({\vartheta }_j\left(y\right)\right)}=0\text{\hspace{0.17em}for\hspace{0.17em}}y\ge y_0$ , under the assumption ${{\displaystyle \int }}^{\infty }{\left(\varsigma \left(\eta \right)\right)}^{-1/a}\text{d}\eta =\infty$ . We consider two cases when $a<c_j$ and $a>c_j$ , where $a$ and $c_j$ are the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result.