Asymptotic property of solutions of a class of third-order differential equations

Abstract:

It has been shown that the equation (*) \[ y''’ + a(t)y'' + b(t)y’ + c(t)y = 0,\] where $a,b$, and $c$ are real-valued continuous functions on $[\alpha ,\infty )$ such that $a(t) \geq 0,b(t) \leq 0$, and $c(t) > 0$, admits at most one solution $y(t)$ (neglecting linear dependence) with the property $y(t)y’(t) < 0,y(t)y''(t) > 0$ for $t \in [\alpha ,\infty )$ and ${\lim _{t \to \infty }}y(t) = 0$, if (*) has an oscillatory solution. Further, sufficient conditions have been obtained so that (*) admits an oscillatory solution.