ON THE GLOBALLY EXPONENTIAL STABILITY OF SOLUTION FOR A CERTAIN CLASS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

This paper gives the description of one general approach to the study of qualitative properties of solutions for nonlinear differential equations. The frequency domain method is an interesting and fruitful technique to determine the stability properties of linear and nonlinear differential equations of various order. Unlike the Lyapunov’s direct method, it does not require the practical construction of a Lyapunov’s function as it has an inbuilt Lyapunov’s function of the type a “quadratic form plus an integral of the non-linear term”. This is the major advantage of the frequency domain method over all other methods in discussing qualitative properties of solution for various order differential equations. The paper is motivated by the paper of “Jipeng Chen”, who used a suitable Lyapunov’s function of the type “quadratic form” only to established sufficient and necessary conditions that guarantee the existence of a solution which is periodic or almost periodic and uniformly ultimately bounded for the equation of the type studied in this paper. We consider the equation of the type (1.1) as in “Jipeng Chen”, and apply the frequency domain method to establishes the necessary and the sufficient conditions that guarantee the existence and unique solution which is periodic, bounded and globally exponentially stable. My approach and results of this study improved on “Afuwape and Omeike” as in the literature to the case where more than two arguments result of the studied equations properties were established. Hence, the results obtained as in the Literature are not the same as in this paper, which implies that the results of this study are essentially new.