On the existence of solutions for nonlinear impulsive periodic viable problems (Q1767440)

The authors study the existence of periodic viable solutions of the following impulsive problem \[ x'(t)\in F(t,x(t))+G(t,x(t)) \;\text{a.e.} \;t\in [0,T]\backslash\{t_{1},\dots,t_{p}\}, \] \[ x(t_{k}^{+})=x(t_{k})+I_{k}(x(t_{k})) \;\text{for any} \;k\in\{1,\dots,p\}, \] \[ x(0)=x(T), \] where \(\Omega\subset \mathbb{R}^{N}, \;N\geq 1\), is a canonical domain, \(F, G:[0,T]\times \bar\Omega\to {\mathcal P}(\mathbb{R}^{N})\) are set-valued maps, \ \(0=t_0<t_1<\dots<t_{p+1}=T, \;I_{k}:\bar\Omega\to \mathbb{R}^{N}\) is an impulse function for \(k\in\{1,\ldots,p\},\) and \(x(t^{+})=\lim_{s\to t^{+}}x(s)\). Existence theorems are given for the particular problem with \(G\equiv 0\).