Ultimate Boundedness Results for a Certain System of Third Order Nonlinear Differential Equations

In this paper, we study the problems of the ultimate boundedness and periodicity of solutions of the third order system of nonlinear differential equations, X⃛ + AẌ + BẊ + H(X) = P(t, X, Ẋ, Ẍ, X⃛), (*) where A, B are real n × n constant symmetric matrices and X ∈ Rn. We obtain some sufficient conditions which ensure that all the solutions of Eq. (*) are ultimately bounded, and we also give some sufficient conditions which guarantee that there exists at least one periodic solution of Eq. (*). Our results revise and improve those results obtained by Afuwape (J. Math. Anal. Appl.97 (1983), 140-150) (which are not applicable to all equations of the general form (*)).