Periodic solutions of Duffing equation with an asymmetric nonlinearity and a deviating argument (Q2015682)

The paper considers a time delay variant of a Duffing type equation introduced by Fučik, namely, \[ x''+ax^+ - bx^- +g(x(t-\tau) = p(t) \] with \(a, b\) constants subject to \(1/\sqrt{a}+1/\sqrt{b}=2/n\), \(n \in N\), \(\tau \in [0, 2 \pi]\) and \(p: \mathbb R \rightarrow \mathbb R\) is continuous and \(2 \pi\)-periodic. The following result is proved. Consider the above equation under the aforementioned basic assumptions. Assume that \(\lim_{x \rightarrow \pm \infty} g(x)\) exists and is finite. Under specific additional assumptions on equation's coefficients and data, existence of at least one \(2 \pi\)-periodic solution is enshured.