Let $(\kappa_t)_{t\ge0}$ be a semigroup of linear, conservative, contractive and nonnegative operators on $C_0(E)$ with $$(\kappa_tf)(x)\xrightarrow{t\to0}f(x)\tag1\;\;\;\text{for all }x\in\mathbb R$$ for all $f\in C_0(\mathbb R)$. Let $(\mathcal D(A),A)$ denote the generator of $(\kappa_t)_{t\ge0}$. Suppose we know that $$\kappa_tf-f=\int_0^t\kappa_s Af\:{\rm d}s\;\;\;\text{for all }f\in\mathcal D(A)\tag2.$$
How can we conclude that $(\kappa_t)_{t\ge0}$ is strongly continuous on $\mathcal D(A)$?
Clearly, $(2)$ again yields $(1)$, but I don't see why the convergence is uniform with respect to $x$.