## Asymptotic behavior of solutions of delayed difference equations.(English)Zbl 1217.39019

Summary: This contribution is devoted to the investigation of the asymptotic behavior of delayed difference equations with an integer delay. We prove that under appropriate conditions there exists at least one solution with its graph staying in a prescribed domain. This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. In the proof of this theorem we show that a good way is to connect two techniques-the so-called retract-type technique and Liapunov-type approach. In the end, we study a special class of delayed discrete equations and we show that there exists a positive and vanishing solution of such equations.

### MSC:

 39A22 Growth, boundedness, comparison of solutions to difference equations 39A30 Stability theory for difference equations
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### References:

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