Asymptotic properties of nonoscillatory solutions of a special half-linear differential equation are investigated. This equation is viewed as a perturbation of the general nonoscillatory equation
One of the main results of the paper reads as follows. Let be a positive solution of (1) such that for large . Suppose that , where , and Then the equation
possesses a pair of linearly independent solutions, which, depending on the value of the parameter , can be expressed by the asymptotic formula
where are roots of a certain quadratic equation and are normalized slowly varying functions. The results of the paper extend, among others, asymptotic formulas given in the paper J. Jaroš, K. Takaŝi and T. Tanigawa [Result. Math. 43, No. 1–2, 129–149 (2003; Zbl 1047.34034)], where (1) reduces to the half-linear Euler differential equation.