On periodic solutions of linear inhomogeneous differential equations with a small perturbation at the derivative

In a Banach space, using branching theory methods, a periodic solution of a linear inhomogeneous differential equation with a small perturbation at the derivative (perturbed equation) is constructed. Under the condition of presence of a complete generalized Jordan set, the uniqueness of this periodic solution is proven. It is shown that when a small parameter is equal to zero and certain conditions are met, the periodic solution of the perturbed equation transforms into the family of periodic solutions of the unperturbed equation. The result is obtained by representing the perturbed equation as an operator equation in Banach space and applying the theory of generalized Jordan sets and modified Lyapunov-Schmidt method. As is known, the latter method reduces the original problem to study of the Lyapunov-Schmidt resolving system in the root subspace. In this case, the resolving system splits into two inhomogeneous systems of linear algebraic equations, that have unique solutions at $\varepsilon \neq 0$, and $2n$-parameter families of real solutions at $\varepsilon=0$, respectively.