zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Periodic solutions of a second order forced sublinear differential equation with delay. (English) Zbl 1097.34050
The authors consider the existence of $2\pi$-periodic solutions to the second-order sublinear differential equation with delay $$ax''(t)+bx(t)+q\bigl (x(t-\tau)\bigr)=p(t),\quad t\in \Bbb R,\tag *$$ where $a,b$ and $\tau>0$ are real constants, the forcing function $p:\Bbb R\to \Bbb R$ is a $2\pi$-periodic continuous function and $g:\Bbb R\to \Bbb R$ is a continuous function. By means of a priori estimation and continuation theorems, the authors obtain criteria for the existence of $2\pi$-periodic solutions of equation (*) under a sublinear condition on the function $g$. The main results of this paper are the following new criteria: (1) If $0<|b|<|a|/ \pi^2$ and if there are constants $\rho>0$, $\beta> 0$ and $\alpha\in [0,1)$ such that $|g(t)|\le\beta|x|^\alpha$ for $|x|>\rho$, then (*) has a $2\pi$-periodic solution. (2) If $b=0$, $a=1$ and if there are constants $\rho>0$ and $\beta\in(0,1/2\pi^2)$ such that $g(x)=-\beta |x|$ for $x\le-\rho$, or $g(x)\le\beta|x|$ for $x\ge\rho$, and $xg(x)> 0$ for $|x|\ge \rho$, then (*) has a $2\pi$-periodic solution. (3) If $b=0$, $a=1$ and there are constants $\rho>0$, $\beta>0$ and $\alpha \in[0,1)$ such that $g(x)\ge-\beta|x|^\alpha$ for $x\le-\rho$, or $g(x) \le\beta|x|$ for $x\ge\rho$, and $xg(x)>0$ for $|x|\ge\rho$, then (*) has a $2\pi$-periodic solution.

34K13Periodic solutions of functional differential equations
Full Text: DOI
[1] Iannacci, R.; Nkashama, M. N.: On periodic solutions of forced second order differential equations with a deviating argument. Lecture notes in math. 1151, 224-232 (1984)
[2] Omari, P.; Zanolin, F.: Boundary value problems for forced nonlinear equations at resonance. Lecture notes in math. 1151, 285-294 (1984)
[3] Huang, X. K.; Xiang, Z. G.: The $2{\pi}$-periodic solution of Duffing equation x”+g(x(t-${\tau}))=p(t)$ with delay. Chinese sci. Bull. 39, No. 3, 201-203 (1994)
[4] Huang, X. K.; Chen, W. D.: The $2{\pi}$-periodic solution of delay Duffing equation x”+cx’+g(x(t-${\tau}))=p(t)$. Progr. natur. Sci. 8, No. 1, 118-121 (1998)
[5] Zhang, Z. Q.; Yu, J. S.: Periodic solution of a kind of Duffing equation. Appl. math. - JCU (Gaoxiao yingyong shuxue xuebao) 13A, No. 4, 389-392 (1998)
[6] Wang, G. Q.; Cheng, S. S.: A priori bounds for periodic solutions of a delay Rayleigh equation. Appl. math. Lett. 12, 41-44 (1999) · Zbl 0980.34068
[7] Ma, S. W.; Wang, Z. C.; Yu, J. S.: Periodic solutions of Duffing equations with delay. Differential equations dynam. Systems 8, No. 3--4, 243-255 (2000) · Zbl 0981.34063
[8] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. Lecture notes in math. 568 (1977) · Zbl 0339.47031
[9] Ding, T. R.: Nonlinear oscillations at a point of resonance. Sci. sinica ser. A. 25, No. 9, 918-931 (1982) · Zbl 0509.34043