Motivated by the recent papers of the first two authors [Math. Comput. Modelling 51, No. 5–6, 658–669 (2010; Zbl 1190.34029), and Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3–4 (A), 2114–2118 (2010; Zbl 1186.34014)], the initial value problem for the second order nonlinear differential equation
is investigated. The functions satisfy certain restrictions which imply, among others, that is a singular point of the investigated equation. One of the main results of the paper gives additional conditions on the nonlinearity under which the singular initial value problem associated with has oscillatory solutions with decreasing amplitudes.