Impulsive integro-differential equations with variable times (Q2763889)

Impulsive integro-differential equations with infinite number of surfaces and variable times are considered. More precisely, the authors provide a criterion for the existence of maximal solution for the the first order impulsive integro-differential equation NEWLINE\[NEWLINE\begin{aligned} \dot x=f(t,x,Tx),\quad &t\not= \tau_{k}(x),\\ \Delta x=\varphi_{k}(x), \quad &t= \tau_{k}(x),\\ x(t_{0}^{+})=x_{0},\quad &\tau_{1}(x_{0}^{+})>t_{0}, \end{aligned}NEWLINE\]NEWLINE where \(f\in C(\mathbb{R}_{+}\times \mathbb{R}^{n}\times \mathbb{R}^{n}, \mathbb{R}^{n})\), \(\varphi_{k}\in C^{1}(\mathbb{R}^{n},\mathbb{R}_{-}^{n})\), \(\mathbb{R}_{-}^{n}:=\{x\in \mathbb{R}^{n}: x<0\}\), \(\tau_{k}(u)\) is an increasing and bounded function of \(u\), \ \(\tau_{k}(u)<\tau_{k+1}(u)\), and \(\lim_{k\to\infty}\tau_{k}(u)=\infty\), \(Tx=\int_{t_{0}}^{t}k(t,s)x(s) ds\), \(k(t,s)\) is a continuous \(n\times n\) matrix for \((t,s)\in \mathbb{R}_{+}\times \mathbb{R}_{+}\) and \(k(t,s)\) is a nondecreasing function of \(s\) for any fixed \(t\).