First order impulsive initial and periodic problems with variable moments (Q1290969)

The periodic impulsive problem \[ y'(t)= f(t,y(t)),\quad\text{a.e. }t\in [0,T], \] \[ y(t^+)= I_i(y(t)),\quad t= \tau_i(y(t)),\quad i= 1,\dots,k,\quad y(0)= y(T),\tag{1} \] is considered. Here, -- \(f:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous and -- \(0<\tau_1(x)<\cdots< \tau_k(x)< \tau_{k+1}(x)\equiv T\), \(x\in\mathbb{R}^n\), are of class \(C^1\). Using a generalization of the Lefschetz fixed point theorem for a class of multivalued maps, sufficient conditions for the existence of a solution to (1) are found. The result is very important and can be generalized in many directions.