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)
Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease. (English) Zbl 0958.45011
The object of the paper is the following nonlinear integral equation $$x(t)=\int^t_{t-\tau} k(t,s)f\bigl(s,x(s) \bigr)ds\quad (t\in \bbfR),\tag 1$$ where $\tau>0$ is a fixed constant and $f(t,x)$ is a real function being periodic in $t$. The equation (1) is a generalization of an integral equation modelling the spread of infectious diseases. Using three fixed point theorems (nonlinear alternative of Leray-Schauder type, the Krasnoselskii fixed point theorem in a cone and a fixed point theorem of {\it R. W. Leggett} and {\it L. R. Williams} [J. Math. Anal. Appl. 76, 91-97 (1980; Zbl 0448.47044)]), the authors established a few interesting existence results for the equation (1). The assumptions of those theorems are rather complicated and too long to be presented here.

45M15Periodic solutions of integral equations
45G10Nonsingular nonlinear integral equations
92C60Medical epidemiology
Full Text: DOI
[1] R.P. Agarwal, D. O’Regan, A fixed point theorem of Leggett Williams type with applications to single and multivalued maps, to appear.
[2] Cooke, K. L.; Kaplan, J. L.: A periodicity threshold theorem for epidemics and population growth. Mat. biosci. 31, 87-104 (1976) · Zbl 0341.92012
[3] Corduneanu, C.: Integral equations and stability of feedback systems. (1973) · Zbl 0273.45001
[4] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045
[5] Leggett, R. W.; Williams, L. R.: A fixed point theorem with application to an infectious disease model. J. math. Anal. appl. 76, 91-97 (1980) · Zbl 0448.47044
[6] M. Meehan, D. O’Regan, Multiple nonnegative solutions to nonlinear integral equations on compact and semiinfinite intervals, to appear.