Oscillation theorems of higher-order nonlinear delay differential equations
Necessary and sufficient conditions for oscillation of a class of higher-order nonlinear delay differential equations of the form: \[ [a(t)x^{(2n-1)}(t)]'+ b(t)\phi[x(t)]p(x'(t))+ \sum q_i(t)g_i[x'(t-\sigma_i(t))] f_i[x(t-\sigma_i(t))]= e(t), \] are given, where \(a\), \(b\), \(q_i\), \(\sigma_i\) \((1\leq i\leq m): I\to\mathbb{R}^+_0= [0,\infty)\), \(I= [t_0,\infty)\); \(p\), \(f_i\) \((1\leq i\leq m)\): \(\mathbb{R}\to\mathbb{R}\); \(\phi\), \(g_i\) \((1\leq i\leq m)\): \(\mathbb{R}\to\mathbb{R}^+\); \(e:I\to\mathbb{R}\). All the functions are continuous and \(xf_i(x)>0\) for \(x\neq 0\) and \(yp(y)>0\) for \(y\neq 0\). Some sufficient conditions and necessary conditions for oscillation of a forced delay differential equation are also given.
higher-order nonlinear delay differential equations
scientific article; zbMATH DE number 707466