This paper provides a complete set of nonresonance conditions for the scalar differential equation
with a strong singularity at the origin and semilinear growth at
. By combining the degree theory and a deep understanding of the relation between the Hill’s equation and the Ermakov-Pinney equation, a technical assumption required in the previous work of P. Yan
and M. Zhang
[Math. Methods Appl. Sci. 26, No. 12, 1067–1074 (2003; Zbl 1031.34040
)] is removed.