Periodic solutions of complex-valued differential equations and systems with periodic coefficients
The paper presents results on the existence of solutions of the periodic problem for some complex-valued systems of ordinary differential equations. At first the system \[ a_i (t) z_i' = |z_i |^q \overline z_i^p + h_i (t,z) \quad (1 \leq i \leq n) \tag{1} \] is considered, where each \(a_i : [0,2 \pi] \to C \backslash \{0\}\) is of \(C^1\)-class and \(2 \pi\)-periodic, \(h_i\) continuous, and \(z = (z_1, \dots, z_n)\). Theorem 1 asserts that if \[ \bigl |h_i (t,z) \bigr |\leq \sum^n_{j = 1} \alpha_{ij} |z_j |^{p + q} + \gamma_i \] where \(i = 1, \dots, n\), \(\alpha_{ij}\) and \(\gamma_i\) are some nonnegative numbers, \(\alpha_{ij}\) are sufficiently small, and \(p + q > 1\) then the periodic problem (2) \(z(0) = z(2 \pi)\) associated to (1) has at least one solution. The proof of this result is based on a variant of Mawhin's theorem on solutions of the coincidence equation. The paper also presents results on the system \[ a_i (t) z_i' = |z_i |^q z_i + h_i (t,z) \quad (1 \leq i \leq n). \tag{3} \] It is assumed that all \(a_i : [0,1 \pi] \to C\) and \(h_i\) are continuous, and the real parts \(\text{Re} a_i\) do not vanish. Using an argument based on the Krasnosel'skij's method of guiding functions it is proved that the problem (2) associated to the system (3) has a solution provided all \(\text{Re} a_i\) have the same sign and some estimates on \(h_i\) hold (Theorem 2). Theorem 3 asserts that the same conclusion holds if \(a_i\) are of \(C^1\)-class, but with possible different signs of \(\text{Re} a_i\) and slightly weaker estimates on the maps \(h_i\). The latter result is proved by a similar argument to the one in the proof of Theorem 1. Several corollaries for the above theorems are presented. Also, some remarks on their relations to nonlinear Ginzburg-Landau and Schrödinger partial differential equations are given.
complex-valued systems of ordinary differential equations
Krasnosel'skij's method of guiding functions
nonlinear Ginzburg-Landau and Schrödinger partial differential equations
scientific article; zbMATH DE number 880560