# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $g\left(·\right):\left(-\infty ,0\right)\cup \left(0,\infty \right)\to R$ is continuous, ${lim}_{|x|\to \infty }g\left(x\right)=0$, ${lim}_{x\to 0±}g\left(x\right)=±\infty ,$ $g\left(x\right)·x>0$ ($\forall \right)x\ne 0$ (in particular, if $g\left(x\right)=1/{x}^{\alpha }$, $\alpha >0\right)$ then the equation: (1) ${u}^{\text{'}\text{'}}+g\left(u\right)=h\left(t\right)$ has a T-periodic solution iff ${\int }_{0}^{T}h\left(t\right)dt\ne 0·$

It is also proved that if $g\left(·\right):\left(0,\infty \right)\to \left(0,\infty \right)$ is continuous and such that: ${lim}_{x\to 0+}g\left(x\right)=+\infty ,{lim}_{x\to \infty }g\left(x\right)=0,{\int }_{0}^{1}g\left(x\right)dx=\infty$ then the equation: (2) ${u}^{\text{'}\text{'}}-g\left(u\right)=h\left(t\right)$ has a T-periodic solution iff ${\int }_{0}^{T}h\left(t\right)dt<0$ and that if ${\int }_{0}^{1}g\left(x\right)dx<\infty$ (in particular, if $g\left(x\right)=1/{x}^{\alpha }$, $0<\alpha <1\right)$ then the equation (2) may not have a periodic solution.

Reviewer: S.Mirica

##### MSC:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34C25 Periodic solutions of ODE
##### Keywords:
subsolutions; resonance; supersolution; truncation