This paper focuses on multiplicity results for the following periodically perturbed autonomous second order retarded functional differential equation with potentially infinite delay: \[ x''(t)=-\alpha x'(t)+ g(x(t))+ \lambda F(t,x_t),\qquad \lambda\geq 0,\tag{1} \] where \(\alpha\geq 0\), \(g: \mathbb{R}\to \mathbb{R}\) is a locally Lipschitz function, and the map \(F: \mathbb{R}\times \mathit{BU}((-\infty,0],\mathbb{R})\to \mathbb{R}\) is \(T\)-periodic in the first variable, and locally Lipschitz in the second one. Here, \(\mathit{BU}((-\infty,0],\mathbb{R})\) denotes the Banach space of all real-valued uniformly continuous bounded functions of \((-\infty,0]\) (equipped with the supremum-norm), and \(x_t\in \mathit{BU}((-\infty,0],\mathbb{R})\) denotes the function \(\theta\mapsto x(t+\theta)\).
Equation (1) can be interpreted as the motion equation of a particle, which is subject to a conservative force plus a possible friction, and a periodic perturbation, which may depend on the whole history of the process. The undelayed (i.e. ODE) case with the absence of friction was studied in [M. Furi et al., Electron. J. Differ. Equ. 2001, Paper No. 36, 9 p. (2001; Zbl 0979.34035)].
One of the main results of the paper states that if \(g\) changes sign at \(n>1\) zeros, then there exists \(\lambda_*>0\), such that for all \(\lambda\in [0,\lambda_*)\), equation (1) has at least \(n\) solutions of period \(T\) that have pairwise no coincident images. Moreover, if \(\alpha\) is positive, then these solutions have mutually disjoint images.
The proofs are based on topological arguments.