On periodic solutions of nonlinear differential equations with singularities. (English) Zbl 0616.34033

Using sub- and supersolution arguments, truncation arguments and previous results of the first author [J. Math. Anal. Appl. 21, 421-425 (1968; Zbl 0155.140)] it is proved that if h(.):R\(\to R\) is T-periodic for some \(T>0\) and if \(g(.): (-\infty,0)\cup (0,\infty)\to R\) is continuous, \(\lim_{| x| \to \infty}g(x)=0\), \(\lim_{x\to 0\pm}g(x)=\pm \infty,\) \(g(x)\cdot x>0\) (\(\forall)x\neq 0\) (in particular, if \(g(x)=1/x^{\alpha}\), \(\alpha >0)\) then the equation: (1) \(u''+g(u)=h(t)\) has a T-periodic solution iff \(\int^{T}_{0}h(t)dt\neq 0.\)
It is also proved that if \(g(.): (0,\infty)\to (0,\infty)\) is continuous and such that: \(\lim_{x\to 0+}g(x)=+\infty,\lim_{x\to \infty}g(x)=0,\int^{1}_{0}g(x)dx=\infty\) then the equation: (2) \(u''- g(u)=h(t)\) has a T-periodic solution iff \(\int^{T}_{0}h(t)dt<0\) and that if \(\int^{1}_{0}g(x)dx<\infty\) (in particular, if \(g(x)=1/x^{\alpha}\), \(0<\alpha <1)\) then the equation (2) may not have a periodic solution.
Reviewer: S.Mirica


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations


Zbl 0155.140
Full Text: DOI


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