COMBINATION OF LIAPUNOV AND RETRACT METHODS IN THE INVESTIGATION OF THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SYSTEMS OF DISCRETE EQUATIONS

This contribution is devoted to a discussion of the asymptotic behavior of solutions of systems of first order nonlinear difference equations. We show that under appropriate conditions there exists at least one solution of the system considered the graph of which stays in a prescribed domain. The domains we work with are the so called polyfacial sets. In literature, retract and Liapunov type approaches are known as excellent asymptotic analysis tools. We present a method which connects both these techniques. Thanks to this, the achieved result can be applied to a substantially wider range of equations. The main result is applied to study a linear system of difference equations as well as to investigate a nonlinear system similar to the discrete scalar equation of Bernoulli’s type. Results are illustrated by detailed examples.