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\(T\)-periodic solutions for some second order differential equations with singularities. (English) Zbl 0761.34031

The authors study the existence of \(T\)-periodic positive solutions of the equation (1) \(x''+f(t,x)=0\) where \(f: \mathbb R\times \mathbb R^ +\to \mathbb R\) is a continuous function, \(T\)-periodic in the first variable, and \(f(t,.)\) has a singularity of repulsive type near the origin. Under the assumption that \(f(t,x)\) lies between two lines of positive slope for large and positive \(x\), they are given in Theorem 1.1 a new non-resonance condition which predicts the existence of one \(T\)-periodic solution of (1). The proof of Theorem 1.1 is based on degree theory together with oscillatory properties of “large” solutions of equation (1). The authors devote this paper to the proof of this result.

MSC:

34C25 Periodic solutions to ordinary differential equations
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[1] DOI: 10.1016/0362-546X(90)90037-H · Zbl 0708.34041
[2] DOI: 10.1090/S0002-9939-1987-0866438-7
[3] Ambrosseti, Analisi Non Lineare (1973)
[4] DOI: 10.1016/0022-1236(89)90078-5 · Zbl 0681.70018
[5] DOI: 10.1016/0022-247X(88)90022-4 · Zbl 0672.34030
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