Double degeneracy in the problem on unbounded branches of forced oscillations (Q951688)

The paper is concerned with the operator equation \[ \mathcal{L}(p;\lambda) x=f(t,x;\lambda) ,\tag{a} \] where \(p=d/dt\) and \(\lambda\in\Lambda\subset\mathbb{R}.\) It is assumed that \(\mathcal{L}\) is a real polynomial in \(p\) and, for any \(\lambda,\) the nonlinearity \(f\) is continuous in \(t\) and \(x\) and \(2\pi\)-periodic in \(t.\) If the polynomial \(\mathcal{L}\) has roots of the form \(ki,\) the set \(\mathfrak{B}_{\varepsilon}\) of all \(2\pi\)-periodic solutions of equation (a) for all \(\left| \lambda-\lambda_{0}\right| \leq\varepsilon\) may be unbounded for any \(\varepsilon.\) Conditions for the set \(\mathfrak{B} _{\varepsilon}\) to be unbounded are derived, and its structure is studied.