Periodic solutions for differential equations at resonance with unbounded nonlinearities (Q1863468)

The authors consider a system of first-order differential equations with a possible delay of the form \[ x'=Bx+F(t,x(t+.))+p(t), \] where the matrix \(B\) has at least one pair of purely imaginary eigenvalues and the functions involved are \(2\pi\)-periodic in \(t\). By using classical techniques on Brouwer degree and Mawhin coincidence degree, it is proved that there exists a \(2\pi\)-periodic solution under a sublinear assumption on \(F\). The main result is applied to a system of coupled Duffing equations with delay and sublinear nonlinearities, improving the results obtained in [\textit{S. Ma, Z. Wang} and \textit{J. Yu}, Nonlinear Anal., Theory Methods Appl. 34, 443-460 (1998; Zbl 0931.34048)].