On the oscillations of mixed neutral equations
The author considers neutral differential equations of odd order of the form \[ (x(t)+ cx(t- h)+ c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*),\tag{1} \] where \(c\), \(c^*\), \(g\), \(g^*\), \(h\), \(h^*\), \(p\) and \(q\) are real constants. It is well-known that a necessary and sufficient condition for oscillation of all solutions of (1) is that the characteristic equation \(z^n(1+ ce^{- hz}+ c^* e^{h^* z})= qe^{- g z}+ pe^{g^* z}\) associated with (1) has no real roots. Since this is not easily verifiable, the author's aim is to obtain sufficient conditions for oscillation of (1) involving the coefficients and the arguments only. A typical result is the following theorem: ``Suppose that \(c^*\), \(g^*\), \(h^*\) and \(p\) are positive constants and \(c\), \(g\), \(h\) and \(q\) are nonnegative constants. Let \[ \Biggl({p\over 1+ c}\Biggr)^{1/n} \Biggl({g^*\over n}\Biggr) e> 1 \] and either \[ q> 0,\;\Biggl({q\over c^*}\Biggr)^{1/n} \Biggl({g+ h^*\over n}\Biggr) e> 1 \] or \[ h^*> g^*,\;\Biggl({p+ q\over c^*}\Biggr)^{1/n} \Biggl({h^*- g^*\over n}\Biggr) e> 1. \] Then the equation \((x(t)+ cx(t- h)- c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*)\) is oscillatory.'' At the end of the paper, the author notes that his results are extendable to more general neutral and nonneutral equations.
neutral differential equations of odd order
scientific article; zbMATH DE number 859211