Solution tube method for impulsive periodic differential inclusions of first order (Q644197)

Consider the periodic impulsive differential inclusion \[ x'(t)\in F(t,x(t)),\quad \text{a.e.}\,\, t\in [0,1]\setminus\{t_{1},\ldots,t_{m}\}, \] \[ \Delta x(t_{i}) =I_{i}(x(t_{i})),\quad i\in \{t_{1},\ldots,t_{m}\}, \tag{1} \] \[ x(0) = x(1). \] The impulses act at the a priori known moments \(0< t_{1}<\cdots< t_{m} < 1\), the jump of the function \(x\) at the time \(t\) is denoted by \(\Delta x(t)= x(t_{+}) - x(t_{-})\), and the discontinuity at the moment \(t_{i}\) is described by the function \(I_{i}:X\to X\), where \(X\) is a real Hilbert space, \(F(t,x)\) represents nonempty, compact convex subsets of \(X\). It is proved that problem (1) has at least one solution if some conditions are fulfilled.