  Functional evolution equations are used in the modeling of numerous physical
processes. In this work, our main tool is perturbation theory of strongly
continuous semigroups. The advantage of this technique is that one can provide
functional evolution equations with the explicit representation formulas of the
solution. First, we introduce a closed form of the fundamental solution of the
evolution equation with a discrete delay using the delayed Dyson-Phillips
series. Then we set up the analytical representation formulas of the classical
solutions of linear homogeneous/non-homogeneous evolution equations with a
constant delay in a Banach space. In the special case, when a strongly
continuous group $\left\lbrace \mathcal{T}(t)\right\rbrace_{t\in \mathbb{R}}$
commutes with a bounded linear operator $A_{1}$, we obtain an elegant formula
for the fundamental solution using the powers of the resolvent operator of
$A_{0}$.
  Furthermore, we consider delay evolution equations with
permutable/non-permutable linear bounded operators and derive crucial results
in terms of non-commutative analysis. Finally, we present an example, in the
context of a one-dimensional heat equation with a discrete delay to demonstrate
the applicability of our theoretical results and give some comparisons with
existing results.
