### 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

#### A. A. Kyashkin^{1}, B. V. Loginov^{2}, P. A. Shamanaev^{3}

Annotation | 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. |
---|---|

Keywords | differential equations in Banach spaces , generalized Jordan sets, Lyapunov-Schmidt resolution system in the root subspace |

^{1}Graduate student of Applied Mathematics,, Differential Equations and Theoretical Mechanics Chair, Mordovian State University after N. P. Ogarev, Saransk; andrej_kjashkin@list.ru.% ^{2}Professor of the Chair "Higher Mathematics" , Ulyanovsk State Technical University, Ulyanovsk; loginov@ulstu.ru% ^{3}Associate Professor of Applied Mathematics, Differential Equations and Theoretical Mechanics Chair,; Mordovian State University after N. P. Ogarev,; Saransk;linebreak korspa@yandex.ru.%

**Citation**: A. A. Kyashkin, B. V. Loginov, P. A. Shamanaev , "[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]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:1 (2016) 45–53 (In Russian)