Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique

Mouataz Billah MESMOULI; Abdelouaheb Ardjouni; Ahcene Djoudi

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2015)

  • Volume: 54, Issue: 1, page 95-108
  • ISSN: 0231-9721

Abstract

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Our paper deals with the following nonlinear neutral differential equation with variable delay d d t D u t ( t ) = p ( t ) - a ( t ) u ( t ) - a ( t ) g ( u ( t - τ ( t ) ) ) - h ( u ( t ) , u ( t - τ ( 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.].

How to cite

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MESMOULI, Mouataz Billah, Ardjouni, Abdelouaheb, and Djoudi, Ahcene. "Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 54.1 (2015): 95-108. <http://eudml.org/doc/271585>.

@article{MESMOULI2015,
abstract = {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.].},
author = {MESMOULI, Mouataz Billah, Ardjouni, Abdelouaheb, Djoudi, Ahcene},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Fixed point theorem; contraction; compactness; neutral differential equation; integral equation; periodic solution; positive solution; stability; Krasnoselskii's theorem; contraction; neutral differential equation; integral equation; periodic solution; fundamental matrix solution; Floquet theory},
language = {eng},
number = {1},
pages = {95-108},
publisher = {Palacký University Olomouc},
title = {Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique},
url = {http://eudml.org/doc/271585},
volume = {54},
year = {2015},
}

TY - JOUR
AU - MESMOULI, Mouataz Billah
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcene
TI - Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2015
PB - Palacký University Olomouc
VL - 54
IS - 1
SP - 95
EP - 108
AB - 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.].
LA - eng
KW - Fixed point theorem; contraction; compactness; neutral differential equation; integral equation; periodic solution; positive solution; stability; Krasnoselskii's theorem; contraction; neutral differential equation; integral equation; periodic solution; fundamental matrix solution; Floquet theory
UR - http://eudml.org/doc/271585
ER -

References

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  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. (2012) Zbl1254.34098MR2988223
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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. (1986) MR0860947
  10. Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Boston, Mass, 1993. (1993) Zbl0777.34002MR1218880
  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. (2012) Zbl1245.34073MR2922961
  12. Smart, D. R., Fixed Points Theorems, Cambridge University Press, Cambridge, 1980. (1980) 
  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. (2013) Zbl1279.34083MR3039158
  14. Zhang, B., 10.1016/j.na.2005.02.081, Nonlinear Anal. 63 (2005), 233–242. (2005) Zbl1159.34348DOI10.1016/j.na.2005.02.081

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