Abstract
In this paper, we study the problems of the ultimate boundedness and periodicity of solutions of the third order system of nonlinear differential equations, X ⃛ + A X ̈ + B X ̇ + H(X) = P(t, X, X ̇ , X ̈ , X ⃛ ), (*) where A, B are real n × n constant symmetric matrices and X ∈ R n . 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 (*)).
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