Positive odd-periodic solutions for a system of second-order ordinary differential equations with derivative terms

In this paper, we study the existence of positive odd 2π-periodic solutions for second-order ordinary differential equations $$ \textstyle\begin{cases} -u''(t)= f(t,u(t),v(t),u'(t)),\quad t\in [0,2\pi ], \\ -v''(t)= g(t,u(t),v(t),v'(t)),\quad t\in [0,2\pi ], \\ u(0)=u(2\pi ),u'(0)=u'(2\pi ), \\ v(0)=v(2\pi ),v'(0)=v'(2\pi ), \end{cases} $$ where $f,g:[0,2\pi ]\times \mathbb{R}^{+}\times \mathbb{R}^{+}\times \mathbb{R}\rightarrow \mathbb{R}^{+}$ are continuous, and $f, g$ are 2π-periodic in t. Under the conditions that nonlinear terms $f(t,x,y,p)$ and $g(t,x,y,q)$ may be superlinear or sublinear growth on $x,y,p$ and q as $|(x,y,p)|\rightarrow 0,~|(x,y,q)|\rightarrow 0$ or $|(x,y,p)|\rightarrow \infty ,~|(x,y,q)|\rightarrow \infty $ . The existence results of positive periodic solutions are obtained, our proof is based on the fixed point index theory in cones. Finally, two examples are given to illustrate the applicability of the conclusions of this paper.