MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2003; 26:1067–1074 (DOI: 10.1002/mma.413) MOS subject classication: 34 M 35; 35 A 20 Higher order non-resonance for dierential equations with singularities Ping Yan † and Meirong Zhang ∗; ‡ Department of Mathematical Sciences; Tsinghua University; Beijing 100084; China SUMMARY In this paper we prove an existence result of positive periodic solutions to second order dierential equations with certain strong repulsive singularities near the origin and with some semilinear growth near innity. Dierent from the nonsingular case, the result in this paper shows that both of the periodic and the antiperiodic eigenvalues play the same role in such an existence result. Copyright ? 2003 John Wiley & Sons, Ltd. KEY WORDS: singular equation; repulsive singularity; positive periodic solution; eigenvalue 1. INTRODUCTION In this paper we are concerned with the existence of (strictly) positive T -periodic solutions of the equation x ′′ + f(t;x)=0 (1) where f : R × R + → R; R + = (0; ∞), is continuous and T -periodic in the rst variable, and f(t;x) exhibits a repulsive singularity near the origin x = 0, i.e., lim x→0+ f(t;x)= -∞. We also assume that f(t;x) grows semilinearly at x = ∞ in the sense that there exist positive T -periodic continuous functions ;  such that (t )6 lim inf x→+∞ f(t;x) x 6 lim sup x→+∞ f(t;x) x 6(t ) (S ∞ ) uniformly in t . ∗ Correspondence to: Meirong Zhang, Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China. † E-mail: yanping99@mails.tsinghua.edu.cn ‡ E-mail: mzhang@math.tsinghua.edu.cn Contract=grant sponsor: National 973 Project Contract=grant sponsor: National NSF Contract=grant sponsor: Ministry of Education of China Copyright ? 2003 John Wiley & Sons, Ltd. Received 6 May 2002