A note on second-order sublinear oscillation theorems (Q1086732)

The author proves integral criteria for the oscillation of solutions to the differential equation \[ (1)\quad u''+a(t)| u|^{\alpha}sgn u=0,\quad (0<\alpha <1), \] where \(a(t)\in C[t_ 0,\infty)\), \(t_ 0>0\). One of these theorems: Assume that for some \(\beta\in [0,\alpha]\) and \(\gamma\in [1,\infty)\) \[ \limsup_{t\to \infty}t^{- \gamma}\int^{t}_{t_ 0}(t-s)^{\gamma}s^{\beta}a(s)ds=\infty, \] then equation (1) is oscillatory. The theorems of this paper generalize Kura's results [\textit{T. Kura}, Proc. Am. Math. Soc. 84, 535-538 (1982; Zbl 0488.34022)].