Periodic Orbits of Nonlinear First-Order General Periodic Boundary Value Problem

Feng Wang, Fang Zhang, Hailong Zhu, Shengjun Li

In this paper, the existence and multiplicity of periodic orbits are obtained for first-order general periodic boundary value problem \[ x'(t)+a(t)x(t)=f(x,t),\qquad tı[0,t], \] \[ x(0)=lpha x(T), \] where $a\colon[0,T]\to[0,+\infty)$ and $f\colon[0,T]\times\mathbb R^+\to\mathbb R$ are continuous functions, $\alpha>0$ and $T>0$ with $\alpha e^{-\int_0^Ta(s)ds}=1$. The proofs are carried out by the use of topological degree theory. We also prove some nonexistence theorems. Our results extend and improve some recent work in the literature.