Existence and uniqueness of periodic solutions for odd-order ordinary differential equations
Volume 100 / 2011
Annales Polonici Mathematici 100 (2011), 105-114
MSC: Primary 34C25; Secondary 34B15.
DOI: 10.4064/ap100-2-1
Abstract
The paper deals with the existence and uniqueness of $2\pi$-periodic solutions for the odd-order ordinary differential equation $$ u^{(2n+1)}=f(t,u,u',\ldots,u^{(2n)}), $$ where $f: \mathbb R\times\mathbb R^{2n+1}\to\mathbb R$ is continuous and $2\pi$-periodic with respect to $t$. Some new conditions on the nonlinearity $f(t,x_0,x_1,\ldots,x_{2n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727–732].
