On the Stability and Ultimate Boundedness of Solutions for Certain Third Order Differential Equations

Abstract

We employ Lyapunov's second method to investigate uniform asymptotic stability, ultimate boundedness and uniform ultimate boundedness of solutions to certain third-order non-linear differential equations. Our results improve some well-known results in the literature.

On the asymptotic behaviour of solutions of certain differential equations of the third order

“…(iii) In the case when f (t, x, y, z) ≡ f (t, x, y) and g(x, y) ≡ g(y) system 1.2 reduces to that discussed by Ademola and Arawomo [2], Swick [24] and Ezeilo [15]. Furthermore, hypotheses and conclusion of Theorem 3.1 coincide with that in [2] Theorem 3.…”

“…As usual, condition for uniqueness of solutions of 1.2 will be assumed andẋ,ẍ, ··· x as elsewhere stand for differentiation with respect to t. Motivation for this work come from the works of Ademola and Arawomo [2], Omeike [19], Qian [20], Swick [23,24] and Tunç [27]. The results obtained in this investigation include and generalize the existing results on third order nonlinear differential equations in the literature.…”

“…(v) When g(x,ẋ) = g(ẋ) and h(x,ẋ) = h(x) system 1.2 reduces to that considered by Ademola and Arawomo [4]. The position of the functionsq(t) andṙ(t) is swaped here.…”

“…Furthermore, hypotheses and conclusion of Theorem 3.1 coincide with that in [2] Theorem 3. Hence our results include and improve [2,15] and [24].…”

“…Notable authors that have contributed immensely to the qualitative behaviour of solutions of third order nonlinear differential equations include Ademola et. al., [1,3,5,6] on uniform stability and boundedness of solutions, Afuwape and Adesina [7], Chukwu [9], Ezeilo [10,11,13], Ezeilo and Tejumola [14], Swick [24], Tejumola [25] and Tunç [26] on boundedness of solutions, Afuwape and Omeike [19], Ogundare [18] and Tunç [28] on convergence of solutions, Omeike [19], Qian [20], Swick [23] and Tunç [27] on asymptotic behaviour of solutions. Most of these works were done with the aid of Lyapunov functions which are either incomplete or contain signum functions.…”