Ultimate boundedness and periodicity results for a certain system Of third-order nonlinear Vector delay differential equations

In the last years, there has been increasing interest in obtaining the sufficient conditions for stability, instability, boundedness, ultimately boundedness, convergence, etc. For instance, in applied sciences some practical problems concerning mechanics, engineering technique fields, economy, control theory, physical sciences and so on are associated with third, fourth and higher order nonlinear differential equations. The problem of the boundedness and stability of solutions of vector differential equations has been widely studied by many authors, who have provided many techniques especially for delay differential equations. In this work a class of third order nonlinear non-autonomous vector delay differential equations is considered by employing the direct technique of Lyapunov as basic tool, where a complete Lyapunov functional is constructed and used to obtain sufficient conditions that guarantee existence of solutions that are periodic, uniformly asymptotically stable, uniformly ultimately bounded and the behavior of solutions at infinity. In addition to being for a more general equation, the obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results.