On oscillation of nonlinear second order differential equation with damping term
The authors study the second-order damped differential equation \[ [r(t)x'(t)]'+p(t)x'(t)+q(t)f(x(t))=0, \quad t\geq t_0, \] where \(r\in C^1([t_0,\infty))\), \(p,q\in C([t_0,\infty))\), \(f\in C((-\infty,\infty))\), \(r\) is a positive function and \(q\) is not identical to zero on any interval \([T,\infty)\) for \(T\) arbitrary large. The function \(f\) is supposed to satisfy the condition \(f(x)/x\geq K>0\) for some positive constant \(K\) and \(x\neq 0\). The authors present several new sufficient conditions which ensure that every continuable proper solution of the equation is oscillatory. The main idea of the proofs is the application of integral averaging technique and the two-parametric function \(H(t,s)\), originally due to \textit{Ch. G. Philos} [Arch. Math. 53, 482-492 (1989; Zbl 0661.34030)] and used extensively in the oscillation theory.
scientific article; zbMATH DE number 2064706