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

Abstract

In this paper, the existence and multiplicity of periodic orbits are obtained for
first-order general periodic boundary value problem
$$
\begin{array}{l}
x'(t)+a(t)x(t)=f(t,x),\ \ t\in [0,T],\\
x(0)=\alpha x(T),
\end{array}
$$
where $a:[0,T]\rightarrow [0,+\infty)$ and $f:[0,T]\times \mathbb{R}^+\rightarrow \mathbb{R}$ are continuous functions, $\alpha>0$ and $T>0$ with $\alpha e^{-\int_0^T a(s)ds}=1$.
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.