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Exponential stability of nonlinear impulsive neutral differential equations with delays. (English) Zbl 1122.34063
The paper deals with the nonlinear impulsive neutral differential equation with time-varying delays of the form $\begin{split} x_i'(t)=&-d_ix_i(t)+\sum_{j=1}^na_{ij}f_{ij}(x_j(t))) +\sum_{j=1}^nb_{ij}g_{ij}(x_j(t-\tau_{ij}(t)))+\\ &+\sum_{j=1}^nc_{ij}h_{ij}(x_j'(t-r_{ij}(t))), \qquad t\geq t_0,\;t\neq t_k,\;i=1,2,\dots ,n, \\ x_i(t)=&I_{ik}(x_1(t^-),\dots ,x_n(t^-)), \qquad t\geq t_0,\;t=t_k,\;i=1,2,\dots ,n, \end{split}\tag{1}$ with the initial conditions $x_i(t_0+s)=\phi_i(s), \qquad -\tau\leq s\leq 0,\;i=1,2,\dots ,n,$ where $0\leq \tau_{ij}(t)\leq \tau, \qquad 0<r_{ij}(t)\leq \tau,$ where $$\tau, a_{ij}, b_{ij}, c_{ij}$$ and $$d_i$$ are constants, $$\tau_{ij}, r_{ij}, f_{ij}, g_{ij}, h_{ij}\in C[\mathbb R, \mathbb R]$$ for $$i,j=1,2,\dots ,n$$. Moreover the initial function $$\phi(s)=(\phi_1(s),\dots ,\phi_n(s))^T\in PC^1$$, the impulsive function $$I_k=(I_{1k}, \dots , I_{nk})^T\in C[\mathbb R^n, \mathbb R^n]$$, and the fixed impulsive moments $$t_k$$ satisfy $$t_1<t_2< \dots$$, $$\lim_{k\to\infty}t_k=\infty$$, $$k=1, 2, \dots$$ . By establishing a singular impulsive delay differential inequality and transforming (1) to a $$2n$$-dimensional singular impulsive delay differential equation, some sufficient conditions are obtained ensuring the global exponential stability in $$PC^1$$ of the zero solution of (1).

##### MSC:
 34K45 Functional-differential equations with impulses 34K40 Neutral functional-differential equations 34K20 Stability theory of functional-differential equations
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##### References:
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