Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point

Mouataz Billah MESMOULI; Abdelouaheb Ardjouni; Ahcene Djoudi

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2016)

  • Volume: 55, Issue: 2, page 129-142
  • ISSN: 0231-9721

Abstract

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In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay x ' t = - a t h x t + d d t Q t , x t - τ t + G t , x t , x t - τ t . The stability of the zero solution of this eqution provided that h 0 = Q t , 0 = G t , 0 , 0 = 0 . The Caratheodory condition is used for the functions Q and G .

How to cite

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MESMOULI, Mouataz Billah, Ardjouni, Abdelouaheb, and Djoudi, Ahcene. "Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 55.2 (2016): 129-142. <http://eudml.org/doc/287913>.

@article{MESMOULI2016,
abstract = {In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay \[ x^\{\prime \}\left( t\right) =-a\left( t\right) h\left( x\left( t\right) \right) +\frac\{d\}\{dt\}Q\left( t,x\left( t-\tau \left( t\right) \right) \right) +G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right) \right) . \] The stability of the zero solution of this eqution provided that $h\left(0\right) =Q\left( t,0\right) =G\left( t,0,0\right) =0$. The Caratheodory condition is used for the functions $Q$ and $G$.},
author = {MESMOULI, Mouataz Billah, Ardjouni, Abdelouaheb, Djoudi, Ahcene},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation; Krasnoselskii-Burton's theorem; large contraction; neutral differential equation; integral equation; periodic solution; non-negative solution},
language = {eng},
number = {2},
pages = {129-142},
publisher = {Palacký University Olomouc},
title = {Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point},
url = {http://eudml.org/doc/287913},
volume = {55},
year = {2016},
}

TY - JOUR
AU - MESMOULI, Mouataz Billah
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcene
TI - Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2016
PB - Palacký University Olomouc
VL - 55
IS - 2
SP - 129
EP - 142
AB - In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay \[ x^{\prime }\left( t\right) =-a\left( t\right) h\left( x\left( t\right) \right) +\frac{d}{dt}Q\left( t,x\left( t-\tau \left( t\right) \right) \right) +G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right) \right) . \] The stability of the zero solution of this eqution provided that $h\left(0\right) =Q\left( t,0\right) =G\left( t,0,0\right) =0$. The Caratheodory condition is used for the functions $Q$ and $G$.
LA - eng
KW - Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation; Krasnoselskii-Burton's theorem; large contraction; neutral differential equation; integral equation; periodic solution; non-negative solution
UR - http://eudml.org/doc/287913
ER -

References

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