On oscillations of third order forced equations (Q1910070)

The paper deals with oscillatory properties of the forced third order equation (E) \(y''+ a(t) y'+ b(t) y'+ c(t) y= f(t)\) under the assumption that the functions \(a\), \(b\), \(c\), \(f\) are sufficiently smooth and \(b- a'\), \(-c\) and \(f\) are nonincreasing for large \(t\). It is shown that \((E)\) admits an oscillatory solution provided \[ \int^\infty \Biggl[ {{2a^2 (t)} \over {27}} - {{a(t) b(t)- a'(t))} \over 3} + c(t) - {2\over {3\sqrt {3}}} \biggl( {{a^2 (t)} \over 3} - (b(t)- a' (t)) \biggr)^{3/2} \Biggr] dt= \infty. \] This statement extends the known result that equation (E), where \(a, c, f>0\), \(b<0\) are real constants, admits an oscillatory solution if and only if \[ {{2a^3} \over {27}} - {{ab} \over 3} + c- {2\over {3\sqrt {3}}} \biggl( {a^2 \over 3} -b \biggr)^{3/2} >0. \]