Geodesic
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, Tome 55 (2016) no. 2, pp. 129-142 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $G$.
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 $G$.
Classification : 34K20, 34K30, 34K40, 47H10
Keywords: Fixed point; stability; delay; stability; nonlinear neutral equation; large contraction mapping; integral equation 
@article{AUPO_2016_55_2_a10,
     author = {MESMOULI, Mouataz Billah and Ardjouni, Abdelouaheb and Djoudi, Ahcene},
     title = {Study of {Stability} in {Nonlinear} {Neutral} {Differential} {Equations} with {Variable} {Delay} {Using} {Krasnoselskii{\textendash}Burton{\textquoteright}s} {Fixed} {Point}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {129--142},
     year = {2016},
     volume = {55},
     number = {2},
     zbl = {06724368},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2016_55_2_a10/}
}
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
SP  - 129
EP  - 142
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/AUPO_2016_55_2_a10/
LA  - en
ID  - AUPO_2016_55_2_a10
ER  - 
%0 Journal Article
%A MESMOULI, Mouataz Billah
%A Ardjouni, Abdelouaheb
%A Djoudi, Ahcene
%T Study of Stability in Nonlinear Neutral Differential Equations with Variable Delay Using Krasnoselskii–Burton’s Fixed Point
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2016
%P 129-142
%V 55
%N 2
%U http://geodesic.mathdoc.fr/item/AUPO_2016_55_2_a10/
%G en
%F AUPO_2016_55_2_a10
MESMOULI, Mouataz Billah; Ardjouni, Abdelouaheb; 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, Tome 55 (2016) no. 2, pp. 129-142. http://geodesic.mathdoc.fr/item/AUPO_2016_55_2_a10/