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. Kyashkin1, B. V. Loginov2, P. A. Shamanaev3
| 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. |
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| Keywords | differential equations in Banach spaces , generalized Jordan sets, Lyapunov-Schmidt resolution system in the root subspace |
1Graduate student of Applied Mathematics,, Differential Equations and Theoretical Mechanics Chair, Mordovian State University after N. P. Ogarev, Saransk; andrej_kjashkin@list.ru.%
2Professor of the Chair "Higher Mathematics" , Ulyanovsk State Technical University, Ulyanovsk; loginov@ulstu.ru%
3Associate 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)
