### 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. Loginov^{1}

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

^{1}Associate Professor of Applied Mathematics, Differential Equations and Theoretical Mechanics Department,; National Research Ogarev Mordovia State University; Saransk;linebreak korspa@yandex.ru. ^{2}Professor 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)