The　problem　of　preservation　of　stability　under　discretization　is　studied.　A　class　of　nonlinear　switched　difference　systems　is　considered.　Systems　of　the　class　appear　as　computational　schemes　for　continuoustime　switched　systems　with　homogeneous　right-hand　sides.　By　using　the　Lyapunov　direct　method,　some　sufficient　conditions　of　the　asymptotic　stability　of　solutions　for　difference　systems　are　obtained.　These　conditions　depend　on　the　information　available　about　the　switching　law.　Three　cases　are　considered.　In　the　first　case,　we　can　guarantee　the　asymptotic　stability　for　any　switching　law,　while　in　the　second　and　in　the　third　ones,　classes　of　switched　signals　are　determined　for　which　the　preservation　of　the　asymptotic　stability　takes　place.　©　Springer-Verlag　Berlin　Heidelberg　2016.