Approximation of the viability kernel

We study recursive inclusionsxn+1ε G(xn). For instance, such systems appear for discrete finite-difference inclusionsxn+1 εGρ(xn) whereGρ:=1+ρF. The discrete viability kernel ofGρ, i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withxn+1 εГρ(xn) whereГρ(x) =x + ρF(x) + (ML/2) ρ2ℬ. Secondly, we show that it can be approached by finite viability kernels associated withxhn+1 ε (Гρ(xhn+1) +α(hℬ) ∩Xh.