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**Akinwale Olutimo**

Department of Mathematics, Lagos State University,

PMB 0001, LASU Post Office, Ojo, Lagos, Nigeria, Senior Lecturer

Dr. and PhD

**Ifeoma Omoko**

Department of Mathematics, Lagos State University,

LASU Post Office, Ojo, Lagos, Nigeria, PhD Student

MSc

We present in this paper the problem of convergence behavior of solutions of certain third-order nonlinear delay differential equations and obtained the sufficient conditions involved under which the solutions of the delay differential equation are convergent. An example is also given to demonstrate the correctness of the proposed approach which improves earlier results on delay differential equations.

- complete Lyapunov function
- Convergence of solutions
- nonlinear delay differential equations of third-order

- Adams, D. O., Olutimo, A. L.;
*Some results on the boundedness of solutions of a certain third order non-autonomous differential equations with delay.*Advanced Studies in Contemporary Mathematics,**29(2)**, 237-249, 2019. http://dx.doi.org/10.107777/ascm2019.29.2.237 - Afuwape, A. U., Omeike, M. O.;
*On the stability and boundedness of solutions of a kind of third order delay differential equations.*Appl. Math. Comput.,**200**, 444-451, 2008 - Afuwape, A. U.;
*Convergence of solutions of certain non-homogeneous third order differential equations.*Kragujevac J. Math.,**31**, 1-12, 2008 - Andres, J.;
*Boundedness results of solutions to the equation x′′′ + ax′′ + g(x)x′ + h(x)= p(t) without the hypothesis h(x)sgn(x) ≥ 0 for |x| > R.*Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. math. Natur,**80(8)**, 533-539, 1987 - Bereketoglu, H., Gyori, I.;
*On the boundedness of the solutions of a third-order nonlinear differential equation.*Dynam. Systems Appl.,**6(2)**, 263-270, 1977 - Chukwu, E. N.;
*On the boundedness of solutions of third order differential equations*. Ann. Mat. Pura Appl.,**155(4)**, 123-149, 1975 - Ezeilo, J. O. C
*.; Further result on the existence of periodic solutions of the equation x′′′ + ψ(x′)x′′ + φ(x)x′ + υ(x, x′, x′′)= p(t, x, x′, x′′) with a bound υ*. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.,**55**, 51-57, 1978 - Ezeilo, J. O. C.; Actes Conference Internat
*. : New properties of the equation x′′′ + ax′′ +bx′ + h(x)= p(t, x, x′, x′′) for certain special values of the incrementary ratio y*Quations differentielles et functionelles non linaires. " Equa-Dff. 73", Brussels/Louvain-la-Neuve, 447-462, Herman, Paris, 1973^{ -1}h(x+y)-h(x). - Li, Q.;
*On the construction of globally asymptotically stable Liapunov's functions of a nonlinear third order systems*. Ann. Differential Equations,**7(1)**, 39-51, 1991 - Olutimo, A. L.;
*On the stability and ultimate boundedness of solutions of certain third order non-autonomous delay differential equations:*In Proceedings of the 14th International Conference: Dynamical Systems-Theory and Applications, Lodz, Poland. Vibration, control and stability of dynamical systems. ISBN 978-83-935312-5-7. (Department of Automation, Biomechanics and Mechatronics), 389-400, 2018 - Olutimo, A. L., Adams, D. O.;
*On the stability and boundedness of solutions of certain non-autonomous delay differential equation of third order.*Applied Mathematics,**7,**457-467, 2016 - Omeike, M. O.;
*New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation*. J. of Inequalities in Pure and Applied mathematics,**9**, Article 15. 8, 2008 - Omeike, M. O.;
*Further results for the solutions of certain third-order differential equations. Nonlinear Analysis.***67**, 3394-3400, 2007 - Omeike, M. O., Olutimo, A. L., Oyetunde, O. O.;
*The boundedness of solutions of certain nonlinear third order ordinary differential equations*. J. Nig. Math. Soc.,**31**, 49-54, 2012 - Oudjedi, L. D., Remili, M.;
*Boundedness and stability in third order nonlinear vector differential equations with bounded delay.*African mathematical Unimand Springer- Verlag GmbH Deutchland cin Teil Von Springer nature 2018 - Qian, C.;
*On global stability of third order nonlinear differential equations*. Nonlinear Anal., Ser. A: Theory methods,**42(4)**, 651-661, 2000 - Reissig, R., Sansone, G., Conti, R.;
*Nonlinear differential equations of higher order.*Translated from the German, Noordhoff International Publishing, Leyden, 1974 - Sadek, A. I.;
*On the stability of solutions of some non-autonomous delay differential equations of third order.*Asymptotic Analysis, 1-7,**43**, 2005 - Swick, K. E
*.; Boundedness and stability for a nonlinear third order differential equation*. Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Fis. mat. Natur.,**56(6)**, 859-865, 1974 - Tunc, C.;
*The boundedness of solutions of nonlinear third order differential equations*. Nonlinear Dynamics and Systems Theory,**10**, 97-102, 2010 - Zhu, Y. F.;
*On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system.*Ann Differential Equation,**8**, 249-259, 1992 - Yao, H., Meng, W.;
*On the stability of solutions of certain non-linear third order delay differential equations.*International Journal of non-linear science,**6**, 230-237, 2008