Ultrametric space – Knowledge and References

Ultrametric space

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Fixed Point Theory For Fuzzy Contractive Mappings

Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021

(b) Let X=[0,2],*=*M,M*(x,y,t)=tt+d*(x,y), where d* is ultrametric. Ultrametric space is metric space where instead of a triangle inequality condition the following is satisfied: d*(x,z) ≤ max{d*(x,y), d*(y,z)}. Then (X,M*,T) is a non-Archimedean fuzzy metric space ([20]). For F(x) = {1,2}, x ∈ X, condition (6.44) is satisfied and by Corollary 3(b) it follows that there exist x ∈ X such that x ∈ Fx.

Formal power series approach to nonlinear systems with additive static feedback

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Journal Information

Published in International Journal of Control, 2023

G. S. Venkatesh, W. Steven Gray

The set is an ultrametric space with the ultrametric where and . For brevity, is written as , and . The ultrametric space is Cauchy complete (Berstel & Reutenauer, 1988). The ultrametric topology on provides a framework to prove the existence of a well-defined feedback product via fixed point theorems as described in Section 5. However, a convergent sequence of series in the ultrametric space , each of which has a well-defined Fliess operator, need not have a well-defined Fliess operator corresponding to the limit. This is demonstrated by the following example.