  The famous 1960s Lumer-Phillips Theorem states that a closed and densely
defined operator $A\colon D(A)\subseteq X\rightarrow X$ on a Banach space $X$
generates a strongly continuous contraction semigroup if and only if $(A,D(A))$
is dissipative and the range of $\lambda-A$ is surjective in $X$ for some
$\lambda>0$. In this paper, we establish a version of this result for
bi-continuous semigroups and apply the latter amongst other examples to the
transport equation as well as to flows on infinite networks.
