| Title:
|
Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique (English) |
| Author:
|
MESMOULI, Mouataz Billah |
| Author:
|
Ardjouni, Abdelouaheb |
| Author:
|
Djoudi, Ahcene |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
95-108 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Our paper deals with the following nonlinear neutral differential equation with variable delay \[ \frac{d}{dt}Du_{t}(t) =p (t)-a(t)u (t)-a(t) g(u(t-\tau (t))) -h (u(t) ,u (t-\tau (t))) . \] By using Krasnoselskii’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. A sufficient condition is established for the positivity of the above equation. Stability results of this equation are analyzed. Our results extend and complement some results obtained in the work [Yuan, Y., Guo, Z.: On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation Abstract and Applied Analysis 2013, ID 175479 (2013), 1–8.]. (English) |
| Keyword:
|
Fixed point theorem |
| Keyword:
|
contraction |
| Keyword:
|
compactness |
| Keyword:
|
neutral differential equation |
| Keyword:
|
integral equation |
| Keyword:
|
periodic solution |
| Keyword:
|
positive solution |
| Keyword:
|
stability |
| MSC:
|
34K20 |
| MSC:
|
34K30 |
| MSC:
|
34K40 |
| MSC:
|
45D05 |
| MSC:
|
45J05 |
| MSC:
|
47H10 |
| idZBL:
|
Zbl 1347.34108 |
| idMR:
|
MR3468603 |
| . |
| Date available:
|
2015-09-01T09:02:29Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144370 |
| . |
| Reference:
|
[1] Ardjouni, A., Djoudi, A.: Fixed points and stability in neutral nonlinear differential equations with variable delays. Nonlinear Anal. 74 (2011), 2062–2070. MR 2781737, 10.1016/j.na.2010.10.050 |
| Reference:
|
[2] Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for a nonlinear neutral differential equation with variable delay. Applied Mathematics E-Notes 2012 (2012), 94–101. Zbl 1254.34098, MR 2988223 |
| Reference:
|
[3] Burton, T. A.: Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem. Nonlinear Studies 9 (2001), 181–190. MR 1898587 |
| Reference:
|
[4] Burton, T. A.: Stability by fixed point theory or Liapunov’s theory: A comparison. Fixed Point Theory 4 (2003), 15–32. MR 2031819 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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[9] Kolmanovskii, V. B., Nosov, V. R.: Stability of functional differential equations. Mathematics in Science and Engineering 180, Academic Press, London, 1986. MR 0860947 |
| Reference:
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| Reference:
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[11] Liu, Z., Li, X., Kang, S., Kwun, Y. C.: Positive periodic solutions for first-order neutral functional differential equations with periodic delays. Abstract and Applied Analysis 2012, ID 185692 (2012), 1–12. Zbl 1245.34073, MR 2922961 |
| Reference:
|
[12] Smart, D. R.: Fixed Points Theorems. Cambridge University Press, Cambridge, 1980. |
| Reference:
|
[13] Yuan, Y., Guo, Z.: On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation. Abstract and Applied Analysis 2013, ID 175479 (2013), 1–8. Zbl 1279.34083, MR 3039158 |
| Reference:
|
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| . |
