Oscillation and asymptotic behavior of odd-order delay differential equations with impulses (Q354845)

\noindent This paper deals with oscillation of solutions for a class of odd-order nonlinear impulsive delay differential equation of the form \[ \begin{aligned} & x^{(n)}(t)+f(t,x(t-\tau ))=0,~t\geq 0,~t\neq t_{k}, \\ & x^{(i)}(t_{k}^{+})=g_{k}^{(i)}(x^{(i)}(t_{k})),~i=0,1,\dots,n-1,~k=1,2,\dots, \\ & x(t)=\phi _{0},~x^{(i)}(t_{0}^{+})=x_{0}^{(i)},~i=1,2,\dots,n-1,~t\in [ t_{0}-\tau ,t_{0}],\end{aligned} \] where \( x^{(0)}(t)=x(t),~n\) is odd,\(~0\leq t_{0}<t_{1}<\dots<t_{k}<\dots\) Several sufficient conditions are obtained for every solution of the given equation which is either oscillatory or eventually tends to zero. Moreover, two examples are given to illustrate the results.