### Existence of a Limiting Regime in the Sense of Demidovich for a Certain Nonlinear Differential Equations of Third Order

#### Abstract

**We extend, in this paper, some known results on the existence of a limiting regime in the sense of Demidovich of some scalar nonlinear differential equations to some third-order nonlinear vector differential equations. Using Lyapunov function as a basic tool and the generalized Theorems of Demidovich, sufficient conditions that guarantee the existence of a limiting regime and almost periodic solutions of certain vector differential equations when the forcing term P is almost periodic in t uniformly in , , and are established. Results obtained are not only new but they also generalize and improve existing results in the relevant literature.**

#### Full Text:

PDF#### References

M.R.M. Rao, Ordinary differential equation, Affliated East-West Private Limited, London, 1980.

L.G. Main, Vibrations and Waves in Physics, Cambridge Univ. Press, Cambridge, 1993.

A.U. Afuwape, Ultimate boundedness results for a certain system of third-order nonlinear differential equations, J. Math. Anal. Appl., 97, 1983, 140 -150.

A.U. Afuwape , Further ultimate boundedness results for a third-order nonlinear system of differential equations, Analisi Funzionale Appl., 6, 1986, 99-100, N.I. 348-360 .

A.U. Afuwape and M.O. Omeike, Further ultimate boundedness solutions of some system of third-order nonlinear ordinary differential equations, Acta univ. Palacki. Olomuc, Fac .rer. nat., Mathematica , 43, 2004, 7-20.

J.O.C. Ezeilo, dimensional extensions of boundedness and stability theorems for some third-order differential equations, J. Math. Anal. Appl., 18, 1967, 395 - 416.

J.O.C. Ezeilo and H.O. Tejumola, Boundedness and periodicity of solutions of a certain system of third-order nonlinear differential equations, Ann. Math. Pura Appl., 4, 1966, 283 - 316.

J.O.C. Ezeilo and H.O. Tejumola, Further results for a system of third-order ordinary differential equations, Atti. Acca Naz, Lincei Reud. Cl. Sci. Fis. Mat. Natur., 58, 1975, 143 - 151.

J.O.C. Ezeilo, Some results for the solutions of a certain system of differential equation, J.Math. Anal. Appl., 6, 1963, 389-393 .

F.W. Meng, Ultimate boundedness results for a certain system of third-order nonlinear differential equations, J. Math. Anal. Appl., Vol. 177, 1993, 496 - 509 .

H.O. Tejumola, On the boundedness and periodicity of solutions of a certain system of third-order nonlinear differential equations, Ann. Math. Pura Appl. IV, Vol. LXXIV, 1966 .

A. Tiryaki, Boundedness and periodicity results for certain system of third order nonlinear differential equations, Indian J.Pure Appl.Math., Vol. 30, No.4, 1999, 361-372.

C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third-order, Nonlinear Analysis, Vol. 70, No.6, 2009, 2232-2236 .

C. Tunc, On the stability and the boundedness of solutions of nonlinear vector differential equations of third-order, Nonlinear Analysis, doc.10:1016/j.na.2008.03.002, 2008.

M.O. Omeike, O.O. Oyetunde and A.L. Olutimo, Bondedness of soltions of certain system of second-order, ordinary differential equations, Acta univ. Palacki. Olomuc, Fac .rer. nat., Mathematica , 53, 2011, 107-115.

A.L. Olutimo, Convergence results for solutions of certain third-order nonlinear vector differential equations, Indian J. Math., 54, No.3, 2012, 299-311 .

A.L. Olutimo, Result on the convergence behavior of solutions of certain system of third order nonlinear differential equations, Int. J. of Modern Nonlinear Theory and Appl., 5, 2016, 48-58 .

A.L. Olutimo and F.O. Akinwole, Stability and boundedness of solutions of certain nonlinear delay differential equations of second-order, J. Math. Comput. Sci., 7, No.3, 2017, 456-467.

M.O. Omeike, Stability and boundedness of solutions of a certain system of third-order nonlinear delay differential equations, Acta univ. Palacki. Olomuc, Fac .rer. nat., Mathematica , 54, N0. 1, 2015, 109-119.

C. Tunc and M. Gozen, Convergence of solutions to a certain vector differential equations of third order, Abstract and Applied Analysis , volume 2014, Article ID 424512, 6 pages.

J.O.C. Ezeilo, New properties of the equation for certain special values of the incrementary ratio , Equations differentielles et functionelles non-lineares edited by P. Janssons, J. Mawhim and N. Rouche, Hermann Publishing, Paris, 1973, pp.447-462.

O.A. Adesina and A.S. Ukpera, On the existence of a limiting regime in the sense of Demidovich for a certain fifth order nonlinear differential equation, Mathematical Analysis, 16, 2009, 193-207.

A.U. Afuwape, On the existence of a limiting regime in the sense of Demidovich for a certain fourth-order nonlinear differential equation, J. Math. Anal. Appl., Vol. 129, No. 2, 1982, 389 -393.

J.O.C. Ezeilo, A generalization of a result of Demidovich on the existence of limiting regime of a system of differential equations, Portugalie Math., 25, 1965, 65-82 .

A.I. Sadek, Stability and boundednes of a kind third-order delay differential system, Applied Math. Letters , 16, 2003, 657-662.

H. Yao and W. Meng, On the stability of solutions of certain nonlinear third-order delay differential equations, Int. J. Nonlinear Science , 6, No. 3, 2008, 230-237.

A.U. Afuwape and M.O. Omeike, On the existence of a limiting regime in the sense of Demidovich for a certain third order nonlinear differential equation, Differential equations and Control Process, Electronic Journal, Vol. 2, 2010, 40-55.

B.P. Demidovich, On the existence of nonlinear system of ordinary differential equations, Amer. Math. Soc. Trans. Ser., Vol. 2, No.18, 1961, 151-161.

### Refbacks

- There are currently no refbacks.

**MAYFEB Journal of Mathematics**

**Toronto, Ontario, Canada**

**MAYFEB TECHNOLOGY DEVELOPMENT**

**ISSN**

**2371-6193**