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)
Convergence of solutions for two delays Volterra integral equations in the critical case. (English) Zbl 1197.45006
Consider the delay Volterra equation $$y(t)=f(t)+\int_{t-\tau}^{t-\delta}k(t,s)g(y(s))ds,\quad t\ge\tau$$ (with initial value $y|_{[0,\tau]}$), where $g$ satisfies a Lipschitz condition with constant $L$. Assuming that $L|k(t,s)|\le\beta(s+\delta)$ with $\beta(t)\le(\tau-\delta)^{-1}-c/t$ it is shown that any two solutions $y,y_*$ of the equation satisfy $|y(t)-y_*(t)|\to0$ as $t\to\infty$, and actually the difference can be estimated by $z(t)-z(t-(\tau-\delta))\to0$ where $z$ is a solution of the auxiliary linear delay differential equation $z'(t)=\beta(t)(z(t-\delta)-z(t-\tau))$. The case is critical in the sense that if, roughly speaking, all inequalities in the assumptions are inverted (in particular, if $L$ denotes the lower Lipschitz constant of $g$) there holds a similar lower bound for $|y(t)-y_*(t)|$.

45G10Nonsingular nonlinear integral equations
45M10Stability theory of integral equations
45D05Volterra integral equations
45M05Asymptotic theory of integral equations
Full Text: DOI
[1] Iannelli, M.; Kostova, T.; Milner, F. A.: A method for numerical integration of age- and size-structured population models, Numer. methods partial differential equations 25, 918-930 (2009) · Zbl 1172.92026 · doi:10.1002/num.20381
[2] Breda, D.; Cusulin, C.; Iannelli, M.; Maset, S.; Vermiglio, R.: Stability analysis of age-structured population equations by pseudospectral differencing methods, J. math. Biol. 54, 701-720 (2007) · Zbl 1114.92054 · doi:10.1007/s00285-006-0064-4
[3] Messina, E.; Russo, E.; Vecchio, A.: A stable numerical method for Volterra integral equations with discontinuous kernel, J. math. Anal. appl. 337, 1383-1393 (2008) · Zbl 1142.65110 · doi:10.1016/j.jmaa.2007.04.059
[4] Messina, E.; Russo, E.; Vecchio, A.: A convolution test equation for double delay integral equations, J. comput. Appl. math. 228, 589-599 (2009) · Zbl 1169.65123 · doi:10.1016/j.cam.2008.03.047
[5] Corduneanu, C.: Integral equations and applications, (1991) · Zbl 0714.45002
[6] Linz, P.: Analytical and numerical methods for Volterra equations, (1985) · Zbl 0566.65094
[7] Diblík, J.; Ružičková, M.: Exponential solutions of equation y?$(t)={\beta}(t)$[y(t-${\delta}$)-y(t-${\tau}$)], J. math. Anal. appl. 294, 273-287 (2004) · Zbl 1058.34099 · doi:10.1016/j.jmaa.2004.02.036
[8] Diblík, J.; Ružičková, M.: Convergence of the solutions of the equation y?$(t)={\beta}(t)$[y(t-${\delta}$)-y(t-${\tau}$)] in the critical case, J. math. Anal. appl. 331, 1361-1370 (2007) · Zbl 1125.34059 · doi:10.1016/j.jmaa.2006.10.001