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)
Necessary and sufficient conditions for ”zero crossing” in integrodifferential equations. (English) Zbl 0735.45007
The authors establish conditions under which all solutions $x(t)$ of the integrodifferential equation $(d/dt)[x(t)-cx(t-\tau)]+a\int\sb 0\sp \infty K(s)x(t-s)ds=0$ vanish at least once on $(-\infty,\infty)$ where $a\in(0,\infty)$, $c\in[0,1)$, $\tau\in[0,\infty)$. If $K$ is ultimately non-increasing and $K\not\equiv0$ on some subinterval of $[0,\infty)$ then it is shown that nontrivial solutions $x(t)$ will possess such zeros (called zero crossings) provided the associated characteristic equation, viz., $\lambda(1-ce\sp{-\lambda\tau})+a\int\sb 0\sp \infty K(s)e\sp{- \lambda s}ds=0$ possesses no real roots.\par\noindent The authors prove that all nontrivial solutions $x(t)$ have zero crossings if $a\int\sb 0\sp \infty K(s)s ds>(1-c)/e$ or $(a+c)\int\sb 0\sp \infty K(s)ds>1/e$. They consider also the logistic integrodifferential equation $dN(t)/dt=rN(t)[1-C\sp{-1}\int\sb 0\sp \infty K(s)N(t-s)ds]$ where $r,C\in(0,\infty)$, $N(0)>0$, $N(s)\ge0$ for $s\in(-\infty,0]$, $K:[0,\infty)\to[0,\infty)$ and $\int\sb 0\sp \infty K(s)ds=1$. If $K$ is ultimately non-increasing and $K\not\equiv0$ on some subinterval of $[0,\infty)$ then all positive solutions $N(t)$ possess ”level crossings”, i.e., $N(t\sp*)=C$ for some $t\sp*\in(- \infty,\infty)$, if $r\int\sb 0\sp \infty K(s)e\sp{\lambda s}ds>\lambda$ for $\lambda\in(0,\infty)$.

45J05Integro-ordinary differential equations
45M15Periodic solutions of integral equations
92D25Population dynamics (general)
Full Text: DOI
[1] O ARINO AND I. GYORI, Necessary and sufficient conditions for oscillation of neutral differential system with several delays, J Differential Equations 81 (1989), 98-105. · Zbl 0691.34054 · doi:10.1016/0022-0396(89)90179-4
[2] N FUKAGAI AND T. KusANO, Oscillation Theory of first order functional differential equations wit deviating arguments, Ann Mat Pura Appl. 136 (1984), 95-117. · Zbl 0552.34062 · doi:10.1007/BF01773379
[3] K GOPALSAMY, Oscillatory properties of systems of first order delay differential inequalities, Pacifi J Math 128 (1987), 299-305 · Zbl 0634.34054 · doi:10.2140/pjm.1987.128.299
[4] K GOPALSAMY AND B. G. ZHANG, Oscillation and nonoscillation in first order neutral differentia equations, J Math Anal. Appl 151 (1990), 42-57 · Zbl 0725.34088 · doi:10.1016/0022-247X(90)90242-8
[5] M K. GRAMMATIKOPOULOS AND I. P. STAVROULAKIS, Necessary and sufficient conditions for oscillatio of neutral equations with deviating arguments, J London Math. Soc (to appear). · Zbl 0719.34133 · doi:10.1112/jlms/s2-41.2.244
[6] M K. GRAMMATIKOPOULOS, Y G. SFICAS AND I. P STAVROULAKIS, Necessary and sufficient condition for oscillations of neutral equations with several coefficients, J Differential Equations 76 (1988), 294-311. · Zbl 0669.34069 · doi:10.1016/0022-0396(88)90077-0
[7] M K GRAMMATIKOPOULOS, E. A. GROVE AND G. LADAS, Oscillation of first order neutral dela differential equations, J Math Anal Appl. 120 (1986), 510-520 · Zbl 0566.34056 · doi:10.1016/0022-247X(86)90172-1
[8] M. K GRAMMATIKOPOULOS, E. A GROVE AND G. LADAS, Oscillation and asymptotic behaviour o neutral differential equations with deviating arguments, Applicable Anal 22 (1986), 1-19 · Zbl 0566.34057 · doi:10.1080/00036818608839602
[9] E A GROVE, G LADAS AND A. MEIMARIDOU, A necessary and sufficient condition for the oscillatio of neutral equations, J Math. Anal. Appl 126 (1987), 341-354. · Zbl 0649.34069 · doi:10.1016/0022-247X(87)90045-X
[10] I. GYORI AND G. LADAS, Oscillations of systems of neutral differential equations, Differential an Integral Equations 1 (1988), 281-287 · Zbl 0723.34057
[11] R JIONG, Oscillations of neutral differential difference equations with several retarded arguments, Sci Sinica A 29, (1986), 1132-1144. · Zbl 0623.34065
[12] V. B KOLMANOVSK AND V. R Nosov, Stability of functional differential equations, Academic Press, New York, 1986
[13] M. R. S. KULENOVIC, G LADAS AND A. MEIMARIDOU, Necessary and sufficient conditions for th oscillation of neutral differential equations, J. Austral. Math. Soc Ser B, 28 (1987), 362-375. · Zbl 0616.34064 · doi:10.1017/S0334270000005452
[14] Y. G. SFICAS AND I. P. STAVROULAKIS, Necessary and sufficient conditions for oscillations of neutra differential equations, J. Math. Anal. Appl. 123 (1987), 494^-507. · Zbl 0631.34074 · doi:10.1016/0022-247X(87)90326-X
[15] I. P. STAVROULAKIS, Oscillations of mixed neutral equations, Hiroshima Math. J. 19 (1989), 441-456 · Zbl 0712.34078