Existence of bounded solutions of discrete delayed equations. (English) Zbl 1065.39006

Aulbach, Bernd (ed.) et al., New progress in difference equations. Proceedings of the 6th international conference on difference equations, Augsburg, Germany July 30–August 3, 2001. Boca Raton, FL: CRC Press (ISBN 0-415-31675-8/hbk). 359-366 (2004).
Summary: A powerful tool for investigation of various asymptotic, boundary-value and qualitative problems in the theory of ordinary differential equations as well as in the theory of delayed differential equations is the retraction method. The development of this method is discussed in the present contribution in the case of one scalar delayed discrete equation of the form \[ \Delta u(k+n)=f\bigl(k,u(k), u(k+1), \dots,u(k+n)\bigr). \] Conditions that guarantee the existence of at least one solution having its graph in a prescribed set are formulated. The proof is based on the idea of a retract principle. In the construction of a retract mapping the property of continuous dependence of solutions on their initial data is used.
For the entire collection see [Zbl 1052.39001].


39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34K05 General theory of functional-differential equations