## 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

### MSC:

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

### Keywords:

subsolutions; resonance; supersolution; truncation

Zbl 0155.140
Full Text:

### References:

  Svatopluk Fučík, Solvability of nonlinear equations and boundary value problems, Mathematics and its Applications, vol. 4, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980. With a foreword by Jean Mawhin.  Robert E. Gaines and Jean L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. · Zbl 0339.47031  A. C. Lazer, On Schauder’s fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl. 21 (1968), 421 – 425. · Zbl 0155.14001  Jean Mawhin, An extension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl. 40 (1972), 20 – 29. · Zbl 0245.34035
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