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Anti-Lyapunov method for systems of discrete equations. (English) Zbl 1065.39008

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.

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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