An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations
The present paper is concerned with the existence and uniqueness of a \(2\pi-\) periodic solution of the differential equation \[ u^{(2n)}(t) +\sum^{n-1}_{j=1} (P_j(t) u^{(j)}(t))^{(j)}+ \sum^n_{j=1} A_ju^{(2j-1)}(t) +(-1)^{(n+1)} \bigtriangledown_uf(t,u)=0, \] where \(P_j(t)\in \mathbb{R}^{l\times l},j=1,\dots,n-1,\; A_j\in \mathbb{R}^{l\times l}, j=1,\dots,n\) are symmetric. The approach is to use a global inversion theorem in the author's paper [J. Math. Anal. Appl. 259, No.~1, 157--167 (2001; Zbl 0994.34028)]. Sufficient conditions and examples are provided.
scientific article; zbMATH DE number 5543775
scientific article