Oscillation criteria for \(n\)th order neutral functional differential equations (Q1331002)

The author considers the \(n\)th order neutral functional differential equation \[ {d^ n\over dt^ n} (x(t)+ cx(t- h)+ c^* x(t+ h^*))+ g x(t- g)+ px(t+ g^*)= 0,\tag{L} \] \(n\) is even, \(c\), \(c^*\), \(h\), \(h^*\), \(p\) and \(g\) are real numbers, \(g\) and \(g^*\) are positive constants. The purpose of this paper is to obtain some easily verifiable sufficient conditions, involving the coefficients and the arguments only. A solution is called oscillatory if it has arbitrarily large zeros. Equation (L) is called oscillatory if all its solutions are oscillatory. Theorem. If \(p,q<0\), \(c^*\leq 0\), \(c\), \(h\), \(h^*\geq 0\), \(g\), \(g^*>0\), \(-c^*\leq c+1\), \(g>h\), \((p/ (1+c))^{1/n}(g^*/n)e>1\), \((q/(1+c))^{1/n}\Bigl({g- h\over n}\Bigr) e>1\), then equation (L) is oscillatory.