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)
Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case. (English) Zbl 1155.34337
Summary: A scalar linear differential equation with time-dependent delay $\dot x(t)= -a(t)x(t-\tau(t))$ is considered, where $t\in I:[t_{0},\infty ), t_0 \in \Bbb R, a: I \to \Bbb R^+:=(0,\infty) $ is a continuous function and $\tau : I \to \Bbb R^+$ is a continuous function such that $t - \tau (t)>t_{0} - \tau (t_{0})$ if $t>t_{0}$. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as $t\rightarrow \infty $ in the critical case in terms of inequalities on $a$ and $\tau $. A generalization of one known final (in a certain sense) result is given for the case of $\tau $ being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.

34K05General theory of functional-differential equations
Full Text: DOI
[1] Diblík, J.: Positive and oscillating solutions of differential equations with delay in critical case, J. comput. Appl. math. 88, 185-202 (1998) · Zbl 0898.34062 · doi:10.1016/S0377-0427(97)00217-3
[2] Diblík, J.; Koksch, N.: Positive solutions of the equation x?$(t)=-c(t)$x(t-${\tau}$) in the critical case, J. math. Anal. appl. 250, 635-659 (2000) · Zbl 0968.34057 · doi:10.1006/jmaa.2000.7008
[3] Domshlak, Y.; Stavroulakis, I. P.: Oscillation of first-order delay differential equations in a critical case, Appl. anal. 61, 359-371 (1996) · Zbl 0882.34069 · doi:10.1080/00036819608840464
[4] Elbert, Á.; Stavroulakis, I. P.: Oscillation and non-oscillation criteria for delay differential equations, Proc. amer. Math. soc. 123, 1503-1510 (1995) · Zbl 0828.34057 · doi:10.2307/2161141
[5] Sljusarchuk, V. E.: The necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with pulse influence in the Banach space, Ukrain. mat. Zh. 51, 98-109 (1999) · Zbl 0938.34024 · doi:10.1007/BF02487410
[6] Agarwal, R. P.; Bohner, M.; Li, Wan-Tong: Nonoscillation and oscillation: theory for for functional differential equations, (2004) · Zbl 1068.34002
[7] Berezansky, L.; Braverman, E.: On exponential stability of linear differential equations with several delays, J. math. Anal. appl. 324, 1336-1355 (2006) · Zbl 1112.34055 · doi:10.1016/j.jmaa.2006.01.022
[8] L. Berezansky, E. Braverman, Positive solutions for a scalar differential equation with several delays, Appl. Math. Lett., in press (doi:10.1016/j.aml./2007.07.017) · Zbl 1146.34325
[9] Berezansky, L.; Domshlak, Yu.; Braverman, E.: On oscillation properties of delay differential equations with positive and negative coefficients, J. math. Anal. appl. 274, 81-101 (2002) · Zbl 1056.34063 · doi:10.1016/S0022-247X(02)00246-9
[10] Čermák, J.: A change of variables in the asymptotic theory of differential equations with unbounded lags, J. comput. Appl. math. 143, 81-93 (2002) · Zbl 1016.34077 · doi:10.1016/S0377-0427(01)00500-3
[11] Čermák, J.: On the related asymptotics of delay differential and difference equations, Dynam. systems appl. 14, 419-429 (2005) · Zbl 1102.34059
[12] Castillo, S.: Asymptotic formulae for solutions of linear functional-differential systems, Funct. differ. Equ. 6, 55-68 (1999) · Zbl 1041.34069
[13] Diblík, J.; Svoboda, Z.: Positive solutions of retarded functional differential equations, Nonlinear anal. 63, e813-e821 (2005) · Zbl 1224.34249 · doi:10.1016/j.na.2005.01.006
[14] Diblík, J.; Svoboda, Z.: Positive solutions of p-type retarded functional differential equations, Nonlinear anal. 64, 1831-1848 (2006) · Zbl 1109.34058 · doi:10.1016/j.na.2005.07.020
[15] Diblík, J.; Ružičková, M.: Exponential solutions of equation y?$(t)={\beta}(t)$[y(t-${\delta}$)-y(t-${\tau}$)], J. math. Anal. appl. 294, No. 1, 273-287 (2004) · Zbl 1058.34099 · doi:10.1016/j.jmaa.2004.02.036
[16] Erbe, L. H.; Kong, Qingkai; Zhang, B. G.: Oscillation theory for functional differential equations, (1995) · Zbl 0821.34067
[17] Györi, I.; Ladas, G.: Oscillation theory of delay differential equations, (1991) · Zbl 0780.34048
[18] Stavroulakis, I. P.: Oscillation criteria for delay and difference equations, Stud. univ. \V{z}ilina math. Ser. 13, No. 1, 161-176 (2003) · Zbl 1057.34084
[19] Hu, Xlaoling; Liu, Guirong; Yan, Jurang: Existence of multiple positive periodic solutions of delayed predator-prey models with functional responses, Comput. math. Appl. 52, 1453-1462 (2006) · Zbl 1128.92047 · doi:10.1016/j.camwa.2006.08.030
[20] Wan, Aying; Jiang, Daqing; Xu, Xiaojie: A new existence theory for positive periodic solutions to functional differential equations, Comput. math. Appl. 47, 1257-1262 (2004) · Zbl 1073.34082 · doi:10.1016/S0898-1221(04)90120-4
[21] Lakshmikantham, V.; Wen, L.; Zhang, B.: Theory of differential equations with unbounded delay, (1994) · Zbl 0823.34069