Periodic Solutions of State-Dependent Abstract Functional Differential Equations with Infinite Delay

In this work, we are concerned in investigating periodic solutions of parametric classes of first- and second-order state-dependent abstract-retarded functional differential equations with infinite delay. To achieve this, we employ the theory of strongly continuous semigroups of operators for first-order equations and the theory of cosine functions of operators for second-order equations. By combining these theories with the concept of topological degree, we successfully establish the existence of periodic solutions for a distinct class of first- and second-order parametric equations. To illustrate our abstract formulation, given its significance in various biological processes, we apply the main result to a class of transport equations with state-dependent delay. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.