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On the oscillations of mixed neutral equations. (English) Zbl 0845.34074

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 * ),(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 -gz +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

p 1+c 1/n g * ne>1

and either

q>0,q c * 1/n g+h * ne>1

or

h * >g * ,p+q c * 1/n h * -g * ne>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.


MSC:
34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory