Fixed set iterations for relaxed Lipschitz multimaps Original Research Article

A dynamical system described by an autonomous differential inclusion with a right-hand side satisfying a relaxed Lipschitz condition, as well as its Euler approximations, are studied. We investigate the asymptotic properties of the solutions and of the attainable sets. It is shown that the system has a strongly flow invariant set, or a “fixed set”, that is, a set such that each trajectory starting from it does not leave it. This set is also an attractor, i.e., it attracts the continuous and the discrete Euler trajectories as the time tends to infinity. We give estimates of the rate of attraction. An algorithm for approximating the fixed set by the attainable sets of the discrete system is also presented.