### 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.

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