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

Hocine Gabsi, Abdelouaheb Ardjouni, Ahcene Djoudi

Abstract


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.

Keywords


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

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