We study \(T\)-periodic solutions of cooperative non-autonomous systems of the form \[u'(t)=f(t,u(t))+F(t), \qquad t\in(0,T),\] in the ordered Banach space \(C_{\mathrm{per}}([0,T];\mathbb{R}^{m})\). Using the explicit periodic resolvent kernel \(K_\lambda\) associated with \(u'+\lambda u=g\), we recast the problem as a fixed-point equation \(u=\mathcal{T}u\) and work in a fully specified Carathéodory framework. More precisely, under assumptions (A1)–(A4) on measurability, regularity, cooperativity and local growth, and a structural condition (H\(_\lambda\)) on the diagonal derivatives of \(f\), we define a monotone, completely continuous operator \(\mathcal{T}\) that leaves invariant the order interval generated by a weak \(T\)-periodic sub- and supersolution. A monotone iteration scheme then yields the existence of weak \(T\)-periodic solutions trapped between the barriers, and we prove the existence of extremal (minimal and maximal) periodic solutions in this interval (Theorem 2). Under an additional Lipschitz condition (A5), we obtain a contraction property for \(\mathcal{T}\), which implies uniqueness and order-stability of the periodic orbit (Proposition 1). As an application, we revisit a water–solute cell-volume model with \(T\)-periodic influx and efflux, derive explicit parameter and bounding conditions ensuring the existence of a strictly positive periodic regime (Theorem 3), and illustrate the qualitative behaviour by a numerical simulation.
