Periodic boundary value problems of impulsive differential equations (Q2723451)

The following impulse periodic boundary value problem is considered: NEWLINE\[NEWLINE\begin{gathered} \dot x(t)=g\bigl(t,x(t) \bigr)+p(t),\quad t\neq t_k,\;k=1,2,\dots,m,\\ x(t^+_k)-x(t_k)= I_k\bigl(t_k,x(t_k) \bigr),\;k=1,2,\dots,m,\tag{1}\\ x(0)=x(T), \end{gathered}NEWLINE\]NEWLINE with \(0=t_0<t_1<t_2 <\cdots<t_m <t_{m+1}=T\), \(T>0\); \(J=[0,T]\), \(g\in C(J\times\mathbb{R}, \mathbb{R})\), \(g(0,u)=g(T,u)\), \(u\in\mathbb{R}\); \(I_k\in C(J \times \mathbb{R},\mathbb{R})\), \(k=1,2,\dots,m\); \(p\in C(J,\mathbb{R})\), \(p(0)=p(T)\). By using results of Mawhin's coincidence degree theory, sufficient conditions for the existence of solutions to (1) are derived.