Anti-Lyapunov method for systems of discrete equations
Sufficient conditions for the boundedness of solutions of the discrete system \[ U_{i}(k+1) = U_{i}(k) + F_{i}(k, \underline{U}(k) \tag{1} \] are considered. By imposing the conditions of Lipschitz, convexity, retract and strict regress-type property the author proves that there exists a solution which belongs to a given open bounded and connected set. Then he gives a variant of the above result using the concept of a polyfacial regular set. He applies his results to the nonlinear system \[ U_{i}(k+1) = \mu_{i}(k)U_{i}(k) + W_{i} (k,\underline{U}(k)) \tag{2} \] and obtains sufficient conditions that the solution \(\underline{U}(k)\) is \(\delta \)-bounded i.e. \( \| \underline{U} \| < \delta \). An example is given. Finally some open problems are discussed.
scientific article; zbMATH DE number 2106070