# 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 class of higher-dimension functional differential equations with impulses. (English) Zbl 1134.34045
The authors study the existence of periodic solutions for the following impulsive system \align x'(t)&=A(t)x(t)+f(t,x_t),\quad t\neq t_k,\ k\in\Bbb Z_+, \\ \Delta x\vert _{t=t_k}&=I(x(t_k)). \endalign As examples they consider the logistic differential equation with several delays, a food-limited delay differential equation, a Lotka-Volterra delay differential competition system.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K45 Functional-differential equations with impulses 34K60 Qualitative investigation and simulation of models
##### Keywords:
Functional-differential equation; applications
Full Text:
##### References:
 [1] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order, J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009 · doi:10.1006/jmaa.1997.5207 [2] Nieto, J. J.: Impulsive resonance periodic problems of first order, Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025 · doi:10.1016/S0893-9659(01)00163-X [3] Qian, D.; Li, X.: Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. math. Anal. appl. 303, 288-303 (2005) · Zbl 1071.34005 · doi:10.1016/j.jmaa.2004.08.034 [4] Yu, J. S.; Zhang, B. G.: Stability theorems for delay differential equations with impulses, J. math. Anal. appl. 199, 162-175 (1996) · Zbl 0853.34068 · doi:10.1006/jmaa.1996.0134 [5] Liu, Q. M.; Dong, S. J.: Periodic solutions for a delayed predator--prey system with dispersal and impulses, E. J. Differential equations 31, 1-14 (2005) · Zbl 1075.34066 · emis:journals/EJDE/Volumes/2005/31/abstr.html [6] Liu, Y. J.; Ge, W. G.: Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear anal. 57, 363-399 (2004) · Zbl 1064.34051 · doi:10.1016/j.na.2004.02.020 [7] Li, W. T.; Huo, H. F.: Existence and global attractivity of positive periodic solutions of functional differential equations with impulses, Nonlinear anal. 59, 857-877 (2004) · Zbl 1061.34059 · doi:10.1016/j.na.2004.07.042 [8] Li, X. Y.; Lin, X. N.; Jiang, D. Q.; Zhang, X. Y.: Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear anal. 62, 683-701 (2005) · Zbl 1084.34071 · doi:10.1016/j.na.2005.04.005 [9] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039 [10] Kuang, Y.: Delay differential equations with application in population dynamics, (1993) · Zbl 0777.34002 [11] Guo, D. J.: Nonlinear functional analysis, (2001) [12] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040 [13] Krasnoselskii, M. A.: Positive solutions of operator equations, (1964) · Zbl 0121.10604 [14] Lan, K.; Jeffry, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities, J. differential equations 148, 407-421 (1998) · Zbl 0909.34013 · doi:10.1006/jdeq.1998.3475