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 discrete equations. Corresponding definition of a point of a type of strict egress for a given set with respect to the system of discrete equations Δu(k) = F (k, u(k)) is given. In an example, it is shown that this definition cannot exactly copy the analogous one, known from the theory of ordinary differential equations. Conditions for existence of at least one solution with graph remaining in a given set are formulated. The proof is based on idea of a retract principle. In construction of a retract mapping, the property of continuous dependence on their initial data is used. Illustrative applications are considered too.