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One can easily see that T t u is a strict contraction with a contractive constant L = 1 − t.
If there is no contractive self-mapping on X not being a strict contraction in the switching law, the above holds if ( X, d ) is just a complete metric space.
Then Ψ is a strict contraction.
Φ 1 is a strict contraction, and.
and hence is a strict contraction whenever is small enough.
A strict contraction (i.e., a contraction A such that (Vert AVert < 1)) is a proper contraction, but a proper contraction is not necessarily a strict contraction.
Obviously, every strict contraction is a proper contraction and every proper contraction is a contraction.
Also, under the condition (mathbb{K}<1), (mathbb{F}) is a strict contraction.
To complete the proof we have to show that is a strict contraction.
its self-commutator ([A^,A]=A^A-AA^) is a strict contraction.
end{aligned} The condition (zeta<1) causes φ to be a strict contraction.
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