The branching of periodic solutions of inhomogeneous linear differential equations with degenerate or identity operator in the derivative and the disturbance in the form of small linear term

In a Banach space existence and uniqueness of periodic solutions of inhomogeneous linear differential equations with degenerate or identity operator in the derivative and a disturbance in the form of small linear term proved by branching theory methods. The article shows that the periodic solution has a pole at the point $ \varepsilon = 0 $ , and if $ \varepsilon = 0 $ the solution goes to $2n$–parameter set of periodic solutions. The result is obtained by applying the theory of generalized Jordan sets, reducing the original problem to the investigation of the Lyapunov-Schmidt resolution system in the root subspace. In this resolution the system is divided into two non-homogeneous systems of linear algebraic equations. These systems have the only solution when $\varepsilon\neq 0$; when $\varepsilon = 0 $ they have $n$-parameter set of solutions, respectively.