New Stability Conditions for the Delayed Lienard Nonlinear Equation via Fixed Point Technique

Hocine Gabsi, Abdelouaheb Ardjouni, Ahcene Djoudi


The class of second order nonlinear neutral differential equations havingthe form\begin{equation*}\ddot{x}+f\left( t,x,\dot{x}\right) \dot{x}+b\left( t\right) g\left( x\left(t-\tau \left( t\right) \right) \right) =0,\text{ for }t\geq t_{0},\end{equation*}%is studied by means of contraction mappings. We give some new conditionsensuring that the zero solution is asymptotically stable. Our results arestrong and do not require conditions that have been indispensable in previousinvestigations such as, $\frac{g\left( x\right) }{x}\geq \beta >0$ and $\lim\frac{g\left( x\right) }{x}$ exists as $x\rightarrow 0$, the delay $\tau\left( t\right) $ is differentiable, the map $t\mapsto t-\tau \left(t\right) $ is strictly increasing (see \cite{pi1}) or the function $b:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is bounded and there exists a function $c:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that $f\left( t,x,y\right) \leq F\left( x,y\right) c\left(t\right) $ for all $t\geq 0$ and $x,y\in\mathbb{R}$. The results obtained improve those of D. Pi \cite{pi1} and are verysignificant because, from practical point of view, it is hard to control thefactors of nuclear reactors that ensure the delay is smoothly changed if wedo not allow dependence between the functions and the time.


xed points, Lienard equation, variable delay, stability, asymptotic stability.

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A. Ardjouni, A. Djoudi, Stability in nonlinear neutral integro-differential equations

with variable delay using xed point theory, J. Appl. Math. Comput. (2014) 44:317{

A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral differential equa-

tions with variable delays, Nonlinear Analysis 74 (2011), 2062-2070.

A. Ardjouni, A. Djoudi, Stability in nonlinear neutral differential with variable de-

lays using xed point theory, Electronic Journal of Qualitative Theory of Differential

Equations, 2011, No. 43, 1{11.

A. Ardjouni, A. Djoudi, Fixed point and stability in neutral nonlinear differential

equations with variable delays, Opuscula Mathematica, Vol. 32, No. 1, 2012, pp.


A. Ardjouni, A. Djoudi, I. Soualhia, Stability for linear neutral integro-differential

equations with variable delays, Electron. J. Differ. Equ., Vol. 2012 (2012), No. 172,


L. C. Becker and T. A. Burton, Stability, xed points and inverse of delays, Proc.

Roy. Soc. Edinburgh 136 A (2006) 245{275.

T. A. Burton, Stability by xed point theory for functional differential equations, Dover

Publications, New York, 2006.

T. A. Burton, Stability by xed point theory or Liapunov's theory: A comparison,

Fixed Point Theory 4 (2003) 15{32.

T.A. Burton, Stability and periodic solutions of ordinary functional differential equa-

tions, Academic Press. NY, 1985.

T. A. Burton, Fixed points, stability, and exact linearization, Nonlinear Analysis, Vol.

(2005), 857{870.

A. Djoudi and R. Khemis, Fixed point techniques and stability for neutral nonlinear

differential equations with unbounded delays, Georgian Mathematical Journal, Vol.

(2006), No. 1, 25{34.

C. H. Jin and J. W. Luo, Stability of an integro-differential equation, Computers and

Mathematics with Applications 57 (2009), 1080-1088.

J. K. Hale, Theory of functional differential equations, Springer Verlag, New York,

NY, USA, 1977.

J. J. Levin and J. A. Nohel, Global asymptotic stability for nonlinear systems of differ-

ential equations and applications to reactor dynamics, Archive for Rational Mechanics

and Analysis, Vol. 5, 1960, pp. 194{211.

M. B. Mesmouli, A. Ardjouni and A. Djoudi, Study of the periodic and nonnega-

tive periodic solutions of functional differential equations via xed points, Azerbaijan

Journal of Mathematics, Vol. 6, No. 2, 2016, pp.70-86.

D. Pi, Study the stability of solutions of functional differential equations via xed

points, Nonlinear Analysis, Vol. 74 (2011), pp. 639{651.

D. Pi, Stability conditions of second order integrodifferential equations with variable

delay, Abstract and Applied Analysis, Vol. 2014 (2014), Article ID371639, 1{11.

D. Pi, Fixed points and stability of a class of integrodifferential equations, Mathemat-

ical Problems in Engineering, Vol. 2014 (2014), Article ID 286214, 1{10.


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