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)
Periodic solutions for a second order nonlinear functional differential equation. (English) Zbl 1151.34056
Define $$L(x(t)):=x''(t)+p(t)x'(t)+q(t)x(t),$$ where $p,q:\Bbb R\to\Bbb R^+$ are continuous $T$-periodic functions with positive average, and $T>0$. The main result of this paper establishes sufficient conditions to ensure the existence of at least one $T$-periodic solution for the second order delay-differential equation $$ L(x(t))=r(t)x'(t-\tau(t))+f(t,x(t),x(t-\tau(t))).\tag{1}$$ Here, $r,\tau:\Bbb R\to\Bbb R$ are continuous and $T$-periodic, and the continuous function $f(t,x,y)$ is $T$-periodic in $t$ for all $(x,y)\in\Bbb R^2$. The Green function for the periodic problem associated to the ordinary differential equation $L(x(t))=\phi(t)$ is used to define a suitable abstract operator whose fixed points are the periodic solutions of (1). Then, a fixed point theorem due to Krasnosel’skii is applied to get the desired existence result. Under an additional condition, this operator is shown to be a contraction, and therefore the $T$-periodic solution is unique.

MSC:
34K13Periodic solutions of functional differential equations
34B27Green functions
WorldCat.org
Full Text: DOI
References:
[1] Freedman, H. I.; Wu, J.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. anal. 23, 689-701 (1992) · Zbl 0764.92016
[2] Kuang, Y.: Delay differential equations with application in population dynamics. (1993) · Zbl 0777.34002
[3] Tang, B.; Kuang, Y.: Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems. Tohoku math. J. 49, 217-239 (1997) · Zbl 0883.34074
[4] Wan, A.; Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu J. Math. 56, 193-202 (2002) · Zbl 1012.34068
[5] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations. Comput. math. Appl. 47, 1257-1262 (2004) · Zbl 1073.34082
[6] Cheng, S.; Zhang, G.: Existence of positive periodic solutions for non-autonomous functional differential equations. Electron. J. Differential equations, No. 59, 1-8 (2001) · Zbl 1003.34059
[7] Zhang, G.; Cheng, S.: Positive periodic solutions of nonautonomous functional differential equations depending on a parameter. Abstr. appl. Anal. 7, 279-286 (2002) · Zbl 1007.34066
[8] Raffoul, Y. N.: Periodic solutions for neutral nonlinear differential equations with functional delay. Electron. J. Differential equations, No. 102, 1-7 (2003) · Zbl 1054.34115
[9] Liu, Y.; Ge, W.: Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients. Tamsui oxf. J. math. Sci. 20, 235-255 (2004) · Zbl 1087.34047
[10] Smart, D. R.: Fixed points theorems. (1980) · Zbl 0427.47036