On the solvability of periodic boundary value problems with impulse (Q878461)

The authors consider an boundary value problem for a first-order differential equation with periodic boundary conditions: \[ x'= f(t, x),\quad t\in [0,N],\quad t\neq t_1,\tag{1} \] \[ x(0)= x(N),\quad 0< N\in\mathbb{R},\tag{2} \] where \(f: [0,N]\times \mathbb{R}^n\to\mathbb{R}^n\) is continuous on \(([0, N]\setminus\{t_1\})\times \mathbb{R}^n\). The impulse at \(t= t_1\) is given by the continuous function \(I_t: \mathbb{R}^n\to\mathbb{R}^n\). By means of the fixed point theorem due to Schäfer, sufficient conditions for the existence of solutions to (1)--(2) are given.