Lower semicontinuity and the theorem of Datko and Pazy

Let {T(t)} t>0 be aC 0-semigroup on a real or complex Banach spaceX and letJ:C +[0,(infinity))-[0, (infinity)] be a lower semicontinuous and nondecreasing functional onC +[0, (infinity)), the positive cone of C[0, (infinity)), satisfyingJ(c 1)= (infinity) for allc>0. We prove the following result: if {T(t)} t>0 is not uniformly exponentially stable, then the set