Exponential stability of equilibria of differential equations with time-dependent delay and non-Lipschitz nonlinearity. (English) Zbl 1206.34096
The authors discuss exponential stability of equilibria of differential equations of the form
\[ \frac{du(t)}{dt}=F(u(t))+G(u(t-\tau(t))),\quad t\geq t_0, \]
\[ u(t)=\phi(t),\quad t\in[t_0-b,t_0]. \]
where \(t_0\geq 0\) and \(b>0\) are constants, \(F\) and \(G\) are nonlinear partially Lipschitz continuous operators from an open subset \(\Omega\) of \(R^n\), \(u(t)\in\Omega\) for \(t\geq t_0\), the delay function \(\tau(t)\) satisfies \(0\leq\tau(t)\leq b\) for \(t\geq t_0\), and \(\phi(\cdot)\in C([t_0-b, t_0],\Omega)\) is an initial function.
Some examples illustrate that the obtained results are improvement and extension of some existing ones.
\[ \frac{du(t)}{dt}=F(u(t))+G(u(t-\tau(t))),\quad t\geq t_0, \]
\[ u(t)=\phi(t),\quad t\in[t_0-b,t_0]. \]
where \(t_0\geq 0\) and \(b>0\) are constants, \(F\) and \(G\) are nonlinear partially Lipschitz continuous operators from an open subset \(\Omega\) of \(R^n\), \(u(t)\in\Omega\) for \(t\geq t_0\), the delay function \(\tau(t)\) satisfies \(0\leq\tau(t)\leq b\) for \(t\geq t_0\), and \(\phi(\cdot)\in C([t_0-b, t_0],\Omega)\) is an initial function.
Some examples illustrate that the obtained results are improvement and extension of some existing ones.
Reviewer: Alexander O. Ignatyev (Donetsk)
MSC:
| 34K20 | Stability theory of functional-differential equations |
Software:
reducedLPReferences:
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