Construction of solutions for delay evolution equations in a Banach space: A delayed Dyson-Phillips series

Functional evolution equations have many applications in modeling many physical processes. First, using the generation theorem, we prove that a fundamental solution of the delay evolution equation is a strongly continuous semigroup of linear operators. Second, we introduce a closed-form representation of a fundamental solution using a delayed Dyson-Phillips series. Then, we establish the analytical representation of the classical solutions of linear homogeneous/nonhomogeneous evolution equations with a discrete delay in a Banach space. In the special case, we consider delay evolution equations with permutable/nonpermutable linear bounded operators and derive crucial results in terms of noncommutative analysis. Furthermore, we prove that a fundamental solution of a functional evolution equation with bounded linear operator coefficients is a uniformly continuous semigroup of linear operators. 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 we give some comparisons with existing results.