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

#### P. A. Shamanaev B. V. Loginov1

Annotation 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. branching of periodic solution, differential equations with delay, generalized Jordan sets, Lyapunov-Schmidt resolution system in the root subspace.

1Associate Professor of   Applied   Mathematics, Differential Equations and Theoretical Mechanics   Department,; National Research Ogarev Mordovia State University; Saransk;linebreak korspa@yandex.ru.

2Professor of "Higher Mathematics" Department , Ulyanovsk State Technical University, Ulyanovsk; loginov@ulstu.ru%

Citation: P. A. Shamanaev B. V. Loginov , "[The branching of periodic solutions of inhomogeneous linear differential equations with a the perturbation in the form of small linear term with delay]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:3 (2016) 61–69 (In Russian)