Exponential stability of nonlinear impulsive neutral differential equations with delays (Q885314)

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).