The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance (Q5932112)

Here, the authors consider the nonlinear system of first-order differential equations with a delay argument: (1) \(x'(t)=Bx(t)+F(x(t-\tau))+p(t)\) with \(x(t)\in \mathbb{R}^2\), \(\tau\in \mathbb{R}\), \(B\in \mathbb{R}^{2\times 2}\), \(F: \mathbb{R}^2\to \mathbb{R}^2\) is bounded and \(p\in C(\mathbb{R},\mathbb{R})\) is \(2\pi\)-periodic. Some sufficient (or necessary) conditions for the existence of \(2\pi\)-periodic solutions to (1), in a resonance, are obtained by using the Brouwer degree theory and a continuation theorem based on Mawhin's coincidence degree. Applications of the main results to Duffing equations with a delay argument are also given.