The author establishes the existence and multiplicity of positive periodic solutions for the first order non-autonomous singular system \[ x_{i}'(t)=-a_{i}(t)x_{i}(t)+\lambda b_{i}(t) f_{i}(x_1(t),\dots,x_n(t)) \qquad i=1,\dots,n\tag{1} \] under the assumptions that

(H1)

\(a_i,b_i\in C(\mathbb{R},[0,\infty))\) are \(\omega\)-periodic functions such that \(\int_0^{\omega}a_i(t)\,dt>0\), \(\int_0^{\omega}b_i\,dt>0\)

(H2)

\(f_i:\mathbb{R}_+^n\setminus\{0\}\to(0,\infty)\) is continuous, \(i=1,\dots,n\).

The main result reads:
{Theorem.} Let (H1), (H2) hold. Assume that \(\lim_{\|u\|\to0}f_i(u)=\infty\) for some \(i=1,\dots,n\). a) If \(\lim_{\|u\|\to\infty}\frac{f_i(u)}{\|u\|}=0, ~i=1,\dots,n\) for all \(\lambda>0\), then (1) has a positive periodic solution. b) If \(\lim_{\|u\|\to\infty}\frac{f_i(u)}{\|u\|}=\infty\) for \(i=1,\dots,n\), then, for all sufficiently small \(\lambda>0\), (1) has two positive periodic solutions. c) There exists a \(\lambda_{0}>0\) such that (1) has a positive periodic solution for \(0<\lambda<\lambda_0\).