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)
A new approach to the existence, nonexistence and uniqueness of positive almost periodic solution for a model of hematopoiesis. (English) Zbl 1225.47072
Summary: Since it is very difficult to obtain the compactness of an almost periodic function set, many classical methods controlled by compact conditions such as Schauder’s fixed point theorem and the coincidence degree cannot be applied to solve almost periodic cases. Therefore, it becomes more complicated to investigate the existence, nonexistence and uniqueness of positive almost periodic solution for a certain model by the traditional methods. In this paper, the authors establish a new fixed point theorem without the compact conditions. As its application, some sufficient conditions for the existence, nonexistence and uniqueness of positive almost periodic solutions for a model of hematopoiesis [{\it L. Glass} and {\it M. C. Mackey}, Science 197, 287--289 (1977; \url{doi:10.1126/science.267326}); cf. also: Ann. New York Acad. Sci., Vol. 316, 214--235 (1979; Zbl 0427.92004)] are obtained. Also, the technique used here is different from the usual methods employed to solve almost periodic cases such as the contraction mapping principle and the Lyapunov functional.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34K14Almost and pseudo-periodic solutions of functional differential equations
92C50Medical applications of mathematical biology
93C15Control systems governed by ODE
Full Text: DOI
[1] Coppel, W. A.: Dichotomies in stability theory, Lecture notes in mathematics 629 (1978) · Zbl 0376.34001
[2] Fink, A. M.: Almost periodic differential equations, Lecture notes in mathematics 377 (1974) · Zbl 0325.34039
[3] Gopalsamy, K.; Trofimchuk, S. I.: Almost periodic solutions of lasota wazewska-type delay differential equation, Journal of mathematical analysis and applications 237, 106-127 (1999) · Zbl 0936.34058 · doi:10.1006/jmaa.1999.6466
[4] Gopalsamy, K.; Weng, P.: Global attractivity and level crossing in model of hematopoiesis, Bulletin of the institute of mathematics, academia sinica 22, 341-360 (1994) · Zbl 0829.34067
[5] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988) · Zbl 0661.47045
[6] Jiang, D.; Wei, J.: Existence of positive periodic solutions for Volterra intergo-differential equations, Acta Mathematica scientia 21B, No. 4, 553-560 (2001) · Zbl 1035.45003
[7] Jiang, D.; Wei, J.: Existence of positive periodic solutions of nonautonomous differential equations with delay, Chinese annals of mathematics 20A, No. 6, 715-720 (1999) · Zbl 0948.34046
[8] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[9] Levitan, B. M.; Zhikov, V. V.: Almost periodic and differential equations, (1978) · Zbl 0414.43008
[10] Liu, Xi-Lan; Li, Wan-Tong: Existence and uniqueness of positive periodic solutions of functional differential equations, Journal of mathematical analysis and applications 293, 28-39 (2004) · Zbl 1057.34094 · doi:10.1016/j.jmaa.2003.12.012
[11] Mackey, M. C.; Glass, L.: Oscillation and chaos in physiological control system, Science 197, 287-289 (1977)
[12] Saker, S. H.: Oscillation and global attractivity in hematopoiesis model with periodic coefficients, Applied mathematics and computation 142, 477-494 (2003) · Zbl 1048.34114 · doi:10.1016/S0096-3003(02)00315-6
[13] Song, Y.; Peng, Y.: Periodic solutions of a nonautonomous periodic model of population with continuous and discrete time, Journal of computational and applied mathematics 188, 256-264 (2006) · Zbl 1100.34039 · doi:10.1016/j.cam.2005.04.017
[14] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations, Computers and mathematics with applications 47, 1257-1262 (2004) · Zbl 1073.34082 · doi:10.1016/S0898-1221(04)90120-4
[15] Wang, H.: Positive periodic solutions of functional differential equations, J. differential equations 202, 354-366 (2004) · Zbl 1064.34052 · doi:10.1016/j.jde.2004.02.018
[16] Wang, X.; Li, Z. X.: Globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients, Applied mathematics and computation 190, 1116-1124 (2007) · Zbl 1131.39010 · doi:10.1016/j.amc.2007.01.106
[17] Wu, X.; Li, J.; Zhou, H.: A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis, Computers and mathematics with applications 54, No. 6, 840-849 (2007) · Zbl 1137.34333 · doi:10.1016/j.camwa.2007.03.004
[18] Xu, B.; Yuan, R.: On the positive almost periodic type solutions for some nonlinear delay integral equations, Journal of mathematical analysis and applications 304, 249-268 (2005) · Zbl 1074.45007 · doi:10.1016/j.jmaa.2004.09.025
[19] Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions, (1975) · Zbl 0304.34051