Show Theorem 8.5 with semigroup methods.

Observe that \(A \int_{0}^{t} P_{s} g d s=P_{t} g-g=\frac{d}{d t} \int_{0}^{t} P_{s} g d s-g\).

Data From Theorem 8.5

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8.5 Theorem. Let (Bt)to be a BMd and g = D(A). Then the unique solution of the Ex. 8.5 inhomogeneous initial value problem (8.7) with g = g(x) satisfying lv(t, x)|