Quasilinearization for the periodic boundary value problem for systems of impulsive differential equations (Q871337)

The authors consider the system of nonlinear impulsive differential equations \[ \begin{aligned} &x'(t) = f(t,x(t)) + g(t,x(t)) \text{ for } t \in [0,T]\setminus\{t_1,\dots,t_p\}, \\ &x(\tau_k + 0) = I_k(x(\tau_k)) + G_k(x(\tau_k))\text{ for } k = 1,\ldots,p,\\ &x(0) = x(T), \end{aligned} \] where \(0 < t_1 < \cdots < t_p < T\); \(f,g : [0,T] \times {\mathbb R}^n \to {\mathbb R}^n\); \(I_k\), \(G_k : {\mathbb R}^n \to {\mathbb R}^n\) are continuous for \(i = 1,\ldots,k\). Sufficient conditions for the existence of a unique solution of the BVP are found. The proofs are based on quasilinearization and lower and upper solutions method.