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

Abstract

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