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

Lazer, A. C.; Solimini, S.

Proc. Am. Math. Soc. 99, 109-114 (1987).

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

Citations:

Zbl 0155.140

References:

[1]	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.
[2]	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
[3]	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
[4]	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