Oscillations in second order differential equations with alternating coefficients (Q1085327)

The authors discuss asymptotic behavior of solutions of \[ (a(t)\psi (x(t))x'(t))'+p(t)x'(t)+q(t)f(x(t))=0. \] It is assumed that functions a, p, q and \(\psi\) are continuous on \([t_ 0,\infty)\), \(t_ 0\geq 0\), f is differentiable and \(x(f)x>0\) for \(x\neq 0\), and both a and \(\psi\) are positive on \([t_ 0,\infty)\). One of the results established is the following: Theorem. Let \(\psi (x)\geq c>0\), and \(f'(x)/\psi (x)\geq \alpha >0\) for \(x\neq 0\). Suppose that there exists a positive differentiable \(\rho\) on \([t_ 0,\infty)\) such that \(\rho\) '(t)p(t)\(\leq \infty\), \(\int ^{\infty}\frac{1}{a(s)\rho (s)}ds<\infty\), and \[ \limsup _{t\to \infty}\int ^{t}_{t_ 0}\frac{1}{a(s)\rho (s)}\int ^{s}_{t_ 0}[\rho (\tau)q(\tau)-\frac{a(\tau)\rho (\tau)}{4\alpha}\{\frac{\rho '(\tau)}{\rho (t)}-\frac{\rho (\tau)}{ca(\tau)}\}^ 2]a\tau ds=\infty. \] Then for every solution x of the equation we have \(\liminf _{t\to \infty}| x(t)| =0\).