Periodic perturbations of a class of functional differential equations
In this paper, the author studies the existence of a connected ``branch'' of periodic solutions of \(T\)-periodic perturbations of a particular class of functional differential equations on differentiable manifolds. Their result is obtained by a combination of degree-theoretic methods and a technique that allows to associate the bounded solutions of the functional equation to bounded solutions of a suitable ordinary differential equation. The author develops their analysis using methods inspired by \textit{M. Furi} et al. [in: Handbook of topological fixed point theory. Berlin: Springer. 741--782 (2005; Zbl 1089.47046); \textit{M. Furi} and \textit{M. Spadini}, Nonlinear Anal., Theory Methods Appl. 29, No. 8, 963--970 (1997; Zbl 0885.34041)]. However, the techniques developed in those papers do not apply directly to perturbations of general functional differential equations but, for the particular form the author observes that the study of periodic solutions can be reduced to the study of the periodic solution of an ordinary differential equation on a higher dimensional manifold. The authors point out that, at the price of some more minor technicalities, the same ``reduction'' strategy adopted in this paper could be applied to the more general case when the periodic perturbation depends also on a retarded term, with delay that might be fixed or functional (possibly infinite).
scientific article; zbMATH DE number 7491618
scientific article