ON THE GLOBALLY EXPONENTIAL STABILITY OF SOLUTION FOR A CERTAIN CLASS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

Main Article Content

EBIENDELE EBOSELE PETER
ASUELINMEN OSORIA

Abstract

This paper gives the description of one general approach to the study of qualitative properties of solutions for nonlinear differential equations. The frequency domain method is an interesting and fruitful technique to determine the stability properties of linear and nonlinear differential equations of various order. Unlike the Lyapunov’s direct method, it does not require the practical construction of a Lyapunov’s function as it has an inbuilt Lyapunov’s function of the type a “quadratic form plus an integral of the non-linear term”. This is the major advantage of the frequency domain method over all other methods in discussing qualitative properties of solution for various order differential equations. The paper is motivated by the paper of “Jipeng Chen”, who used a suitable Lyapunov’s function of the type “quadratic form” only to established sufficient and necessary conditions that guarantee the existence of a solution which is periodic or almost periodic and uniformly ultimately bounded for the equation of the type studied in this paper. We consider the equation of the type (1.1) as in “Jipeng Chen”, and apply the frequency domain method to establishes the necessary and the sufficient conditions that guarantee the existence and unique solution which is periodic, bounded and globally exponentially stable. My approach and results of this study improved on “Afuwape and Omeike” as in the literature to the case where more than two arguments result of the studied equations properties were established. Hence, the results obtained as in the Literature are not the same as in this paper, which implies that the results of this study are essentially new.   

Keywords:
Globally exponentially stable, third order, differential equation, properties and lyapunov’s function

Article Details

How to Cite
PETER, E. E., & OSORIA, A. (2021). ON THE GLOBALLY EXPONENTIAL STABILITY OF SOLUTION FOR A CERTAIN CLASS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS. Asian Journal of Current Research, 6(3), 11-17. Retrieved from https://www.ikprress.org/index.php/AJOCR/article/view/7073
Section
Original Research Article

References

Abou- El – Ela AMA, Sadek AI, Mahmoud AM. Stability and boundedness of solutions of a certaine third order nonlinear delay differential. ICGST – ACSE Journal. 2009;9(1):9 -15.

Ademola AT, Arawomo PO. Uniform stability and boundedness of solutions on delay differential equations of the third order. Mathematical Journal of Okayama University. 2013;55:157 -166.

Adesina OA. Results on the qualitative behavior of Solutions for a class of third order nonlinear Differential equations. AIP Conference Proceedings. 2014;1637(1):5 -12,

Afuwape AU, Omeike MO. On the boundedness of solutions of a kind of third order delay differential equations. Appl. Math. Compat. 2008;200:444-451.

Barbalat I, Halanay A. Applications of the frequency domain method to force nonlinear Oscillations. Math. Nathr. 1970;44:165–179.

Bernis F, Peletier A. Two Problems from draining flows involving third order ordinary differential equations. SIAM J. Math. Anal. 1996;515–527.

Bal Z. Existence of solutions for some third – order boundary – value Problems, Electron. J. Differential Equations. 2009;256.

Ebiendele EP, Adamu B. Application of Navier -Stokes equations to solve Fluid flow problems. Journal of Advances in Mathematics and Computer Science. 2020;35(8):101– 114.

Ebiendele EP. On the stable and unstable state of a certain class of delay differential equations. Archives of Applied Science Research. 2017;9(3):35–40.

Ebiendele EP. On Uniform boundedness and stability of solutions for predator – Prey models for delay nonlinear differential equations. Asian Journal of Mathematics and Computer Research. 2018;25(4):238–248.

Ebiendele EP. New criterion that guarantee sufficient conditions for globally asymptotically stable periodic solutions of nonlinear differential equations with delay. Journal of Advances in Mathematics and Computer Science. 2019;31(5):1 -10.

Elakhe AO, Aliu kA, Ebiendele EP. An Explicit one – step method of an order eight rational integrator. IOSR Journal of Mathematics. 2020;16(6):1 – 09.

Jipeng Chen. Existence of the uniformly ultimate bounded solutions and periodic solutions. Annals of Differential Equations. 1998;4:103–109.

Lewis A, Olutimo Samuel, Iyase A, Hilary I. Okagbue. convergence behavior of solutions of a kind of third – order nonlinear differential equations. Advances in Differential Equations and Control Processes. 2020;23(1)1–19.

Omeike MO. New results on the stability of Solution of some non – autonomous delay differential equations. Control Processes. 2010;1:18 -29.

Tunc C. On Asymptotic Stability of Solutions to third order nonlinear differential equations with retarded argument. Appl. Anal. 2007;11(3 -4):515–527.

Tunc C. A new boundedness result to nonlinear differential equations of third order with finite lag. Commun. Appl. Anal. 2009;13(1): 1–10.

Yacubovich AU. Solution of some matrix inequalities occurring in the theory of automatic control. Soviet Math. Dokl. 1962;4:1304–1307,

Zhou XY, Song X, Shi X. A differential equation model of HIV infection of CD_4^+ T- cell with cure rate, J. Math. Anal. Appl. 2008;342: 1342 -1355.