Oscillation criteria for \(n\)th-order forced functional-differential equations
Using the method of general means, the authors present some new oscillation criteria for forced functional-differential equations of the form \[ x^{(n)}(t)+\sum_{i=1}^{n-1}a_ix^{(i)}(t)+q(t)f(x(g(t)))=e(t), \] where \(a_i\) are real constants, \(q(t),\;f(x),\;e(t)\), and \(g(t)\) are real continuous functions, \(xf(x)>0\) for \(x\neq 0\), and \(\lim_{t\to\infty}g(t)=\infty\). The same technique is used to establish some oscillation criteria for the forced neutral differential equation of the form \[ (x(t)+\delta(t)x(t-\tau))^{(n)}+ \sum_{i=1}^{n-1}a_i(x(t)+\delta(t)x(t-\tau))^{(i)}+ f(t,x(g(t)),x'(\sigma(t)))=e(t), \] and the neutral equation of the form \[ \begin{multlined} (x(t)+\delta x(t-\tau))^{(n)}+ \sum_{i=1}^{n-1}a_i(x(t)+\delta x(t-\tau))^{(i)}+q(t)x(t)+ b(t)x(t-\tau)\\ =e(t)+c(t)f_1(x(t))+d(t)f_2(x(t-\tau)), \end{multlined} \] where \(xf_1(x)>0\) and \(xf_2(x)>0\) for \(x\neq 0, n\geq 1\), \(\tau, \delta\) are nonnegative constants, \(q(t)>0\), \(b(t)>0\), \(c(t)>0\), and \(d(t)>0\).
\(n\)th-order forced functional-differential equations
scientific article; zbMATH DE number 1915728