Oscillation theorems for damped-forced \(n\)th order nonlinear differential equations with deviating arguments
Consider the equations (1) \(x^{(n)}(t)+q(t)f(x[g(t)])h(x^{(n- 1)}[\sigma (t)])=0\), n is even, and \[ (2)\quad x^{(n)}(t)+q(t)f(x[\sigma (t)])h(x^{(n-1)}[\sigma (t)])=e(t), \] n is even, where g,q,\(\sigma\) : [t\({}_ 0,\infty)\to R-(-\infty,\infty)\), f,h: \(R\to R\) are continuous, q(t)\(\geq 0\) and not identically zero on any ray of the form \([t^*,\infty)\) for some \(t^*\geq t_ 0\), g(t)\(\to \infty\), \(\sigma\) (t)\(\to \infty\) as \(t\to \infty\), and \(h(u)>0\) for \(u\neq 0\), e: [t\({}_ 0,\infty)\to R\) is continuous and oscillatory. Assume that \(xf(x)>0\) for \(x\neq 0\), \(f(xy)\geq Kf(x)f(y)\) on (0,\(\infty)\), and \(-f(-xy)>Kf(x)f(y)\) on (-\(\infty,0)\), where K is a positive constant. The authors give sufficient conditions for every solution of equation (1) to be oscillatory and sufficient conditions for every solution x of equation (2) to be either oscillatory or \(x^{(n-1)}(t)\to 0\) as \(t\to \infty\).
scientific article; zbMATH DE number 4109177
scientific article