Suppose K is a continuous matrix-valued function such thatand let X be the matrix-valued solution of the resolvent problemthen the following are equivalent: (a) There are β > 0, c > 0 such that |K(t)| ≤ ce -βt , t ≥ 0. (b) There are α > 0, c 0 > 0 such that |X(t)| ≤ c 0 e -αt , |X (t)| ≤ c 0 e -αt , |X (t)| ≤ c 0 e -αt , t ≥ 0.
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Appleby et al. (2004) studied this question.
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