| Title:
|
Existence and positivity of solutions for a nonlinear periodic differential equation (English) |
| Author:
|
Yankson, Ernest |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
261-270 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the existence and positivity of solutions of a highly nonlinear periodic differential equation. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ a modification of Krasnoselskii’s fixed point theorem introduced by T. A. Burton ([4], Theorem 3) to show the existence and positivity of solutions of the equation. (English) |
| Keyword:
|
fixed point |
| Keyword:
|
large contraction |
| Keyword:
|
periodic solution |
| Keyword:
|
positive solution |
| MSC:
|
34A12 |
| MSC:
|
34A37 |
| MSC:
|
39A05 |
| idMR:
|
MR3007609 |
| DOI:
|
10.5817/AM2012-4-261 |
| . |
| Date available:
|
2012-12-17T13:50:30Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143101 |
| . |
| Reference:
|
[1] Burton, T. A.: Integral equations, implicit relations and fixed points.Proc. Amer. Math. Soc. 124 (1996), 2383–2390. MR 1346965, 10.1090/S0002-9939-96-03533-2 |
| Reference:
|
[2] Burton, T. A.: A fixed point theorem of Krasnoselskii.Appl. Math. Lett. 11 (1998), 85–88. Zbl 1127.47318, MR 1490385, 10.1016/S0893-9659(97)00138-9 |
| Reference:
|
[3] Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem.Nonlinear Stud. 9 (2002), 181–190. Zbl 1084.47522, MR 1898587 |
| Reference:
|
[4] Burton, T.A.: Stability by fixed point theory for functional differential equations.Mineola, NY, Dover Publications, Inc., 2006. Zbl 1160.34001, MR 2281958 |
| Reference:
|
[5] Chen, F. D.: Positive periodic solutions of neutral Lotka-Volterra system with feedback control.Appl. Math. Comput. 162 (3) (2005), 1279–1302. Zbl 1125.93031, MR 2113969, 10.1016/j.amc.2004.03.009 |
| Reference:
|
[6] Chen, F. D., Shi, J. L.: Periodicity in a nonlinear predator-prey system with state dependent delays.Acta Math. Appl. Sinica (English Ser.) 21 (1) (2005), 49–60. Zbl 1096.34050, MR 2123604, 10.1007/s10255-005-0214-2 |
| Reference:
|
[7] Curtain, R. F., Pritchard, A. J.: Functional analysis in modern applied mathematics.Mathematics in Science and Engineering, Vol. 132. London–New York, Academic Press, 1977. Zbl 0448.46002, MR 0479787 |
| Reference:
|
[8] Elkadeky, W. K., El-Sayed, A. M.: Caratheodory theorem for a nonlocal problem of the differential equation $x^{\prime }=f(t,x^{\prime })$.Alex. J. Math. 1 (2) (2010), 8–14. |
| Reference:
|
[9] Fan, M., Wang, K.: Global periodic solutions of a generalized $n$-species Gilpin–Ayalacompetition model.Comput. Math. Appl. 40 (10–11) (2000), 1141–1151. MR 1784658, 10.1016/S0898-1221(00)00228-5 |
| Reference:
|
[10] Hafsia, D., Ahcene, D.: Periodic solutions for nonlinear differential equation with functional delay.Georgian Math. J. 15 (4) (2008), 635–642. Zbl 1171.47061, MR 2494962 |
| Reference:
|
[11] Hafsia, D., Ahcene, D.: Existence of periodic solutions for neutral nonlinear differential equations with variable delay.Electron. J. Differential Equations 127 (2010), 1–8. Zbl 1203.34110, MR 2685037 |
| Reference:
|
[12] Kaufmann, E. R.: A nonlinear neutral periodic differential equation.Electron. J. Differential Equations 88 (2010), 1–8. Zbl 1200.34094, MR 2680291 |
| Reference:
|
[13] Kun, L. Y.: Periodic solution of a periodic neutral delay equation.J. Math. Anal. Appl. 214 (1997), 11–21. Zbl 0894.34075, MR 1645495, 10.1006/jmaa.1997.5576 |
| Reference:
|
[14] Raffoul, Y. N.: Periodic solutions for neutral nonlinear differential equations with functional delays.Electron. J. Differential Equations 102 (2003), 1–7. MR 2011575 |
| Reference:
|
[15] Raffoul, Y. N.: Positive periodic solutions in neutral nonlinear differential equations.Electron. J. Qual. Theory Differ. Equ. 16 (2007), 1–10. Zbl 1182.34091, MR 2336604 |
| . |
