The branching of periodic solutions of inhomogeneous linear differential equations with a the perturbation in the form of small linear term with delay

In a Banach space by branching theory methods existence and uniqueness of periodic solutions of inhomogeneous linear differential equations with degenerate or identity operator in the derivative and a perturbation in the form of small linear term with delay is proved. The article shows that the periodic solution has a pole at the point $ \varepsilon = 0 $ , and if $ \varepsilon = 0 $ it goes to $2n$–parameter set of periodic solutions. The result is obtained by applying the theory of generalized Jordan sets, that reduces the original problem to the investigation of the Lyapunov-Schmidt resolution system in the root subspace. This resolution system is a non-homogeneous system of linear algebraic equations, which at $ \varepsilon \neq 0 $ has a unique solution, and at a value of $ \varepsilon = 0 $ goes to $2n$-parameter family of solutions.