Discrete retract principle for systems of discrete equations

A power tool for investigation of various problems of ordinary differential equations as well as delayed differential equations is a retraction method . The development of this method is discussed here in the case of system di screte equations. Corresponding definition of a point of a type of strict e gress for a given set with respect to the system of discrete equations Delt au(k) = F(k, u(k)) is given. In an example, it is shown that this definitio n cannot exactly copy the analogous one, known from the theory of ordinary differential equations. Conditions for existence of at least one solution w ith graph remaining in a given set are formulated. The proof is based on id ea of a retract principle. In construction of a retract mapping, the proper ty of continuous dependence of solutions on their initial data. is used. Il lustrative applications are considered too. (C) 2001 Elsevier Science Ltd. All rights reserved.