Set-Induced Stability Results for Delay Difference Equations

This chapter focuses on the relation between stability of delay difference equations (DDEs) and the existence of \(\mathcal{D}\)-contractive sets. Such sets are of importance as they provide a region of attraction, which is difficult to obtain for delay systems. Firstly, it is established that a DDE admits a \(\mathcal{D}\)-contractive set if and only if it admits a Lyapunov-Razumikhin function. However, it is also shown that there exist stable DDEs that do not admit a \(\mathcal{D}\)-contractive set. Therefore, secondly, further necessary conditions for the existence of a \(\mathcal{D}\)-contractive set are established. These necessary conditions provide a first step towards the derivation of a notion of asymptotic stability for DDEs which is equivalent to the existence of a \(\mathcal{D}\)-contractive set.

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