Existence and uniqueness of periodic solution for a class of differential systems
The authors use a global inverse function theorem to prove the existence and uniqueness of periodic solutions for differential systems of the type \[ x_k' = f_k (t,x_1,x_2,\ldots,x_n) \] where \(f_k\) is continuous and \(2\pi\)-periodic in \(t\) and \(\partial f_k / \partial x_j\) is continuous and bounded, for each \(1 \leq k,j \leq n.\) Moreover, the following assumption is supposed: for any continuous vector function \(x(t) = (x_1(t),\ldots,x_n(t)), \;t \in [0,2\pi],\) define the matrix \(A(x) = (a_{ij}(x)),\) where \[ a_{ij}(x) = \int_0^{2\pi} \frac{\partial f_i}{\partial x_j} \;(t,x_1 (t),\ldots,x_n (t)) \;dt \] Then \(A(x)\) is nonsingular and the eigenvalues \(\lambda_k = \lambda_k (x)\) of \(A(x)\) satisfy: For those \(\{ \lambda_j \} \), where \(R(\lambda_j) \neq 0,\) there exists a positive constant \(\delta\) such that \(| R(\lambda_j) | \geq \delta\) (\(R(\lambda_j)\) is the real part of \(\lambda_j\)), and for those \(\{ \lambda_k \} \) with \(R(\lambda_k)=0,\) there exist nonnegative integers \(\{ m_k \}\) and positive constants \(\{a_k\}, \{b_k\}\) such that \[ 2\pi m_k < a_k \leq | \lambda_k (x) | \leq b_k < 2\pi (m_k+1). \] Some applications to even-order differential scalar equations are shown.
scientific article; zbMATH DE number 5083256