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Let (T t)) ((tgeq0)) be an exponentially stable (C_{0} -semigroup, for (forallnuin(0} -semigroup, by the deforition oforallnuin 0we have (exists Mgeq1), biglVert T(t)bigrVert leq Me^{-nu t},quad tgeq0.
Combining this fact with the normality of cone K in X, we have (exists M_{1}>0), (forall v_{n-1}in G_{0}), biglVert f_{1}bigl t,v_{n-1}(t bigr bigrVert leq biglVert f_{1}bigl t,v_{0}(t bigr bigrVert +N_{0}biglVert f_{1}bigl t,w_{0}(t) bigr -f_{1}bigr -f_{{0}(t)bigr)bigrVert leq M_{1}bigl t
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com