This paper deals with existence and the number of periodic solutions of the first order system \[ u'_i(t)=-a_i(t)u_i(t)+\lambda b_i(t)f_i(\vec u(t)),\quad i=1,\,\dots,\,n\eqno(1) \] where \(\vec u=(u_1,\,\dots,\,u_n)\in{\mathbb R}^n\), \(a_i\) and \(b_i\) are continuous \(T\)-periodic functions with positive mean value in a period, \(f_i: {\mathbb R}^n\setminus0\to]0,\infty[\) are continuous functions and \(\lambda\) is a positive parameter.
In addition it is assumed that \(f_i(\vec u)\to +\infty\) as \(\|\vec u\|\to0\) and \(\frac{f_i(\vec u)}{\|\vec u\|}\to+\infty\) as \(\|\vec u\|\to\infty\).
The main result (showing that (1) mimics the behaviour of the autonomous single equation, which can be reduced to that of a real function) is the following: there exists \(\lambda^*>0\) such that (1) has at least 2, at least 1, or no \(T\)-periodic solutions according to whether \(0<\lambda<\lambda^*\), \(\lambda=\lambda^*\) or \(\lambda>\lambda^*\) respectively.
The proof makes use of lower and upper solutions and the usual degree-theoretic argument.
Related results had been given by H. Wang [J. Differ. Equations 249, No. 12, 2986–3002 (2010; Zbl 1364.34032); Appl. Math. Comput. 218, No. 5, 1605–1610 (2011; Zbl 1239.34043)].