Functional boundary value problem for first order impulsive differential equations at variable times. (Q1432752)

By using the method of upper and lower solutions, the authors prove some existence results for extremal solutions of the following functional boundary value problems for first-order impulsive differential equations at variable times of the form \[ \begin{aligned} &x'(t)=f(t,x(t)), \quad t\neq \tau(x(t)), \;t\in J:=[0,T],\\ &\Delta x(t)=I(x(t)), \quad t=\tau(x(t)),\\ &B(x(0),x)=0, \end{aligned} \] where \(f\in C(J\times \mathbb{R}, \mathbb{R}),\) \(I\in C^1(\mathbb{R}, \mathbb{R}),\) \(\tau n \in C^1(\mathbb{R},\mathbb{R}),\) and \(B\in C(J\times PC(J, \mathbb{R}),\mathbb{R})\). \(B\) includes initial conditions if \(B(a, \zeta) = B(a) = a -x_{0}\) for all \(a \in \mathbb{R}\), and a periodic condition if \( B(a, \zeta) = \zeta(T) -a\) for all \((a,\zeta) \in \mathbb{R} \times PC (J, \mathbb{R})\).