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**A. T. Ademola**

Department of Mathematics, Faculty of Science

Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

**S. Moyo**

Institute for Systems Science & Research and Postgraduate Support Directorate

Durban University of Technology, Durban 4000, South Africa

**B. S. Ogundare**

Department of Mathematics, Faculty of Science

Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

**M. O. Ogundiran**

Department of Mathematics, Faculty of Science

Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

**O. A. Adesina**

Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

In this work we consider a class of third order delay differential equations, where the nonlinear functions, especially the first two restoring terms, are sum of n multiple deviating arguments, the forcing term depends explicitly on the independent variable t for all i in [1..n], the last restoring term has variable coefficient, and deviating arguments vary for all i. By employing the direct technique of Lyapunov, where a complete Lyapunov functional is constructed and used, we obtain sufficient conditions that guarantee the existence of solutions which are periodic, uniformly asymptotically stable, uniformly ultimately bounded. The behaviour of solutions as t tends to infinity is studied. The obtained results are new and include many recent results in the literature. Finally, two examples are given to show the feasibility of our results.

- periodic solutions
- Third order nonlinear differential equation
- uniform stability
- uniform ultimate boundedness

- Ademola, A. T., Arawomo, P. O.;
*Asymptotic behaviour of solutions of third order nonlinear differential equations,*Acta Univ. Sapientiae, Mathematica, 3(2), 197 - 211 (2011) - Ademola, A. T., Arawomo, P. O.;
*Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order.*Math. J. Okayama Univ. 55, 157 - 166 (2013) - Ademola, A. T., Arawomo, P. O., Ogunlaran, M. O., Oyekan, E. A.;
*Uniform stability, boundedness and asymptotic behaviour of solutions of some third order nonlinear delay differential equations.*Differential Equations and Control Processes. N 4, 43 - 66 (2013) - Ademola, A. T., Arawomo, P. O.;
*Boundedness and asymptotic behaviour of solutions of a nonlinear differential equation of the third order.*Afr. Mat. 23:DOI 10. 1007/s13370-011-0034-x 261 - 271 (2012) - Ademola, A. T., Arawomo, P. O.;
*On the asymptotic behaviour of solutions of certain differential equations of the third order*. Proyecciones Journal of Mathematics. 33(1), 111 - 132 (2014) - Ademola, A. T., Arawomo, P. O.;
*Stability, boundedness and asymptotic behaviour of solutions of certain nonlinear differential equations of the third order.*Kragujevac Journal of Mathematics. 35(3), 431 - 445 (2011) - Ademola, A. T.;
*Existence and uniqueness of a periodic solution to certain third order nonlinear delay differential equation with multiple deviating arguments.*Acta Univ. Sapientiae Mathematica. 5(2), 113 - 131 (2013) - Ademola, A. T.;
*Stability, boundedness and uniqueness of solutions to certain third order stochastic delay differential equations*. Differential Equations and Control Processes. N 2, 25 - 50 (2017) - Ademola, A. T., Mahmoud, A. M.;
*On some qualitative properties of solutions to certain third order vector differential equations with multiple constant deviating arguments*. International Journal of Nonlinear Science, 26 (3), 180 - 192 (2018) - Ademola, T. A., Ogundiran, M. O.;
*On the existence and uniqueness of solutions of a generalized Lipschitz ordinary differential equations.*Ife Journal of Science, 9(2), 241 - 246 (2007) - Ademola, T. A., Ogundiran, M. O., Arawomo, P. O., Adesina, O. A.;
*Boundedness results for a certain third order nonlinear differential equation.*Applied Mathematics and Computation 216 3044 - 3049 (2010) - Ademola, A. T., Ogundare, B. S., Ogundiran, M. O., Adesina, O. A.;
*Stability, boundedness and existence of periodic solutions to certain third order delay differential equations with multiple deviating arguments.*International Journal of Differential Equations. Volume 2015, Article ID 213935, 12 pages (2015) - Ademola, A. T., Ogundiran, M. O., Arawomo, P. O.;
*Stability, boundedness and existence of periodic solutions to certain third order nonlinear differential equations.*Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica. 54(1) 5 - 18, (2015) - Adesina, O. A.;
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*Uniform ultimate boundedness of solutions of third order nonlinear delay differential equations.*An. Stiint. Univ. Al. I. Cuza Iasi, Ser. Noua, Mat., Tomul LVI. f. 2 DOI: 10. 2478/v10157-010-0026-4. 363 - 372 (2010) - Remili, M., Beldjerd, D.;
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*Stability and boundedness of the solutions of non autonomous third order differential equations with delay.*Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica. 53(2), 139 - 147 (2014) - Tunç, C.;
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*Stability and boundedness for a kind of non-autonomous differential equations with constant delay.*Appl. Math. Inf. Sci. 7(1), 355-361 (2013) - Tunç, C.;
*Stability and boundedness in differential systems of third order with variable delay Stability and boundedness in differential systems of third order with variable delay*. Proyecciones 35(3), 317-338 (2016) - Tunç, C
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*On the qualitative behaviors of nonlinear functional differential systems of third order.*Advances in nonlinear analysis via the concept of measure of noncompactness, 421 - 439, Springer, Singapore, (2017) - Tunç, C.;
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*Uniformly boundedness of a class of non linear differential equations of third order with multiple deviating arguments*. Cubo 14(3) , 63 - 69 (2012) - Tunç, C ., Gö zen, M.;
*Stability and uniform boundedness in multi delay functional differential equations of third order.*Abstract and Applied Analysis. Vol. 2013, Article ID 248717, 7, (2013) - Tunç, C., Mohammed, S. A.;
*On the qualitative properties of differential equations of third order with retarded argument*. Proyecciones. 33(3), 325 - 347 (2014) - Yao, H., Wang, J.;
*Globally asymptotic stability of a kind of third order delay differential system*. International Journal of Nonlinear Science. 10(1), 82 - 87 (2010) - Yoshizawa, T.;
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