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We study the existence of periodic solutions for second-order neutral functional differential systems with operator varying in time and delays. An interesting issue is that the coefficient matrix of the neutral difference operator is not a constant matrix, which contradicts the available literature data. Some properties of the neutral difference operator variable as a function of time and the results on the existence of periodic solutions of the second-order neutral functional differential systems are obtained. Moreover, numerical simulations are performed to illustrate the theoretical results., 1. Introduction In recent years, the problem of existence of periodic solutions to functional differential systems has been extensively investigated (see [1, 2, 4, 5, 7-12] and the references therein). [...]

In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: x ″ + f (x ′) + g (t , x) = s , x (0) − x (T) = 0 = x ′ (0) − x ′ (T) , where f : ℝ → ℝ is a continuous function with f (0) = 0 , function g : ℝ / T ℤ × ℝ + → ℝ is continuous with an attractive singularity at the origin, and s is a constant. We consider the case where the friction term f satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function g does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained. [ABSTRACT FROM AUTHOR]