| Title:
|
Systems of nonlinear delay integral equations modelling population growth in a periodic environment (English) |
| Author:
|
Cañada, A. |
| Author:
|
Zertiti, A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
4 |
| Year:
|
1994 |
| Pages:
|
633-644 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type $$ x(t) = \int_{t-\tau _1}^t f(s,x(s),y(s))\,ds $$ $$ y(t) = \int_{t-\tau _2}^t g(s,x(s),y(s))\,ds $$ which model population growth in a periodic environment when there is an interaction between two species. For the proofs, we develop an adequate method of sub-supersolutions which provides, in some cases, an iterative scheme converging to the solution. (English) |
| Keyword:
|
nonlinear integral equations |
| Keyword:
|
monotone methods |
| Keyword:
|
population dynamics |
| Keyword:
|
positive solutions |
| MSC:
|
34K15 |
| MSC:
|
45G10 |
| MSC:
|
45G15 |
| MSC:
|
45M15 |
| MSC:
|
45M20 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 0816.45002 |
| idMR:
|
MR1321234 |
| . |
| Date available:
|
2009-01-08T18:14:00Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118705 |
| . |
| Reference:
|
[1] Amann H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces.Siam Review 18 (1976), 620-709. Zbl 0345.47044, MR 0415432 |
| Reference:
|
[2] Ca nada A.: Method of upper and lower solutions for nonlinear integral equations and an application to an infectious disease model.in ``Dynamics of Infinite Dimensional Systems'', S.N. Chow and J.K. Hale editors, Springer-Verlag, Berlin, Heidelberg, 1987, 39-44. MR 0921896 |
| Reference:
|
[3] Ca nada A., Zertiti A.: Method of upper and lower solutions for nonlinear delay integral equations modelling epidemics and population growth.$M^3AS$, Math. Models and Methods in Applied Sciences 4 (1994), 107-120. MR 1259204 |
| Reference:
|
[4] Ca nada A., Zertiti A.: Topological methods in the study of positive solutions for some nonlinear delay integral equations.to appear in J. Nonlinear Analysis. MR 1305767 |
| Reference:
|
[5] Cooke K.L., Kaplan J.L.: A periodic threshold theorem for epidemics and population growth.Math. Biosciences 31 (1976), 87-104. MR 0682251 |
| Reference:
|
[6] Guo D., Lakshmikantham V.: Positive solutions of integral equations arising in infectious diseases.J. Math. Anal. Appl. 134 (1988), 1-8. MR 0958849 |
| Reference:
|
[7] Krasnoselskii M.A.: Positive solutions of operator equations.P. Noordhoff, Groningen, The Netherlands, 1964. MR 0181881 |
| Reference:
|
[8] Nussbaum R.D.: A periodicity threshold theorem for some nonlinear integral equations.Siam J. Math. Anal. 9 (1978), 356-376. Zbl 0385.45007, MR 0477924 |
| Reference:
|
[9] Smith H.L.: On periodic solutions of a delay integral equations modelling epidemics.J. Math. Biology 4 (1977), 69-80. MR 0504059 |
| Reference:
|
[10] Torrejon R.: A note on a nonlinear integral equation from the theory of epidemics.J. Nonl. Anal. 14 (1990), 483-488. Zbl 0695.45008, MR 1044076 |
| . |
