Akinwale L. Olutimo^*

Show more
Department of Mathematics, Lagos State University, Ojo, Nigeria.
Show less

Abstract: Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.

Keywords: Nonlinear Differential Equations, Third Order, Convergence of Solutions, Lyapunov Method

Cite this paper: Olutimo, A. (2016) Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations. International Journal of Modern Nonlinear Theory and Application, 5, 48-58. doi: 10.4236/ijmnta.2016.51005.

References

[1]   Reissig, R., Sansome, G. and Conti, R. (1974) Nonlinear Differential Equations of Higher Order. Noordhoff, Groningen.

[2]   Abou-El Ela, A.M.A. (1985) Boundedness of the Solutions of Certain Third Order Vector Differential Equations. Annals of Differential Equations, 1, 127-139.

[3]   Afuwape, A.U. (1983) Ultimate Boundedness Results for a Certain System of Third-Order Nonlinear Differential Equations. Journal of Mathematical Analysis and Applications, 97, 140-150.
http://dx.doi.org/10.1016/0022-247X(83)90243-3

[4]   Afuwape, A.U. (1986) Further Ultimate Boundedness Results for a Third-Order Nonlinear System of Differential Equations. Analisi Funzionale e Applicazioni, 6, 348-360.

[5]   Chukwu, E.N. (1975) On the Boundedness of Solutions of Third-Order Differental Equations. Annali di Matematica Pura ed Applicata, 4, 123-149.
http://dx.doi.org/10.1007/BF02417013

[6]   Ezeilo, J.O.C. (1967) n-Dimensional Extensions of Boundedness and Stability Theorems for Some Third-Order Differential Equations. Journal of Mathematical Analysis and Applications, 18, 395-416.
http://dx.doi.org/10.1016/0022-247X(67)90035-2

[7]   Ezeilo, J.O.C. and Tejumola, H.O. (1966) Boundedness and Periodicity of Solutions of a Certain System of Third-Order Nonlinear Differential Equations. Annali di Matematica Pura ed Applicata, 4, 283-316.
http://dx.doi.org/10.1007/BF02416460

[8]   Meng, F.W. (1993) Ultimate Boundedness Results for a Certain System of Third-Order Nonlinear Differential Equations. Journal of Mathematical Analysis and Applications, 177, 496-509.
http://dx.doi.org/10.1006/jmaa.1993.1273

[9]   Omeike, M.O. (2014) Stability and Boundedness of Solutions of Nonlinear Vector Differential Equations of Third-Order. Archivum Mathematicum, 50, 101-106.
http://dx.doi.org/10.5817/AM2014-2-101

[10]   Omeike, M.O. and Afuwape, A.U. (2010) New Result on the Ultimate Boundedness of Solutions of Certain Third Order Vector Differential Equations. Acta Universitatis Palackianae Olomucensis Facultas Rerum Naturalium Mathematica, 49, 55-61.

[11]   Tiryaki, A. (1999) Boundedness and Periodicity Results for Certain System of Third Order Nonlinear Differential Equations. Indian Journal of Pure and Applied Mathematics, 30, 361-372.

[12]   Tunc, C. (1999) On the Boundedness and Periodicity of Solutions of a Certain Vector Differential Equations of Third-Order. Applied Mathematics and Mechanics, 20, 163-170.

[13]   Tunc, C. (2009) On the Stability and Boundedness of Solutions of Nonlinear Vector Differential Equations of Third-Order. Nonlinear Analysis, 70, 2232-2236.
http://dx.doi.org/10.1016/j.na.2008.03.002

[14]   Tunc, C. and Ates, M. (2006) Stability and Boundedness Results for Solutions of Certain Third-Order Nonlinear Vector Differential Equations. Nonlinear Dynamics, 45, 273-281.
http://dx.doi.org/10.1007/s11071-006-1437-3

[15]   Tunc, C. and Mohammed, S.A. (2014) On the Qualitative Properties of Differential Equations with Retarded Argument. Proyeccious Journal of mathematics, 33, 325-347.

[16]   Afuwape, A.U. (1983) On the Convergence Solutions of Certain System of Nonlinear Third-Order Differential Equations. A Quarterly Journal of Pure and Applied mathematics, 57, 225-271.

[17]   Afuwape, A.U. and Omeike, M.O. (2005) Convergence of Solutions of Certain System of Third Order Nonlinear Ordinary Differential Equations. Annals of Differential Equations, 21, 533-540.

[18]   Afuwape, A.U. (2009) Convergence of Solutions of Some Third Order Systems of Non-Linear Ordinary Differential Equations. Analele Stiintice ale Universitatii A.I. 1. Cuza din Lasisenie Noua. Mathematica, 55.

[19]   Tunc, C. and Gozen, M. (2014) Convergence of Solutions to a Certain Vector Differential Equations of Third Order. Abstract and Applied Analysis, 2014, Article ID: 424512.

[20]   Olutimo, A.L. (2012) Convergence Results for Solutions of Certain Third-Order Nonlinear Vector Differential Equations. Indian Journal of Mathematics, 54, 299-311.