Sizar Abid Mohammed and Cemil Tunc.  Qualitative Analysis of Nonlinear Retarded Differential Equations of Second Order.  Dynamic Systems and Applications 29 (2020) No. 1, 53-70

https://doi.org/10.46719/dsa20202914

ABSTRACT.
By this research paper, we construct Lyapunov–Krasovskii functional and Lyapunov
function, respectively, for non-linear functional differential equations (FDEs) of second order, in other words, for delay differential equations (DDEs) of second order, with constant retardations. By that auxliary functional and function we are able to establish five new results on the global stability and eventually uniform boundedness of solutions, square integrability of the first derivative of solutions, the existence of the periodic solutions and the existence and the uniqueness of the stationary oscillation. Here, the arguments of discussion are based only upon the second method of Lyapunov–Krasovskii functional and Lyapunov function approaches by applying some well-known theorems for the qualitative analysis of solutions of FDEs. The results introduced improve and extend the main results of the papers ([18, 53]) obtained for FDEs before. In particular case of FDEs, illustrative examples and their graphs are given to highlight the applicability of the results introduced.

AMS (MOS) Subject Classification: 34K12, 34K13, 34K20.
Key Words and Phrases: FDEs; second order; multiple constant retardations; global stability; eventually uniform boundedness; existence of periodic solutions; square integrability; stationary oscillation.