CUBO A Mathematical Journal Vol.14, N^o 03, (63–69). October 2012

 

Uniformly boundedness of a class of non-linear differential equations of third order with multiple deviating arguments

 

Cemil Tunç , Hilmi Ergören

Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080, Van, TURKEY email: cemtunc@yahoo.com, hergoren@yahoo.com

---

ABSTRACT

This paper deals with a certain third-order non-linear differential equation with multiple deviating arguments. Some sufficient conditions are set up for all solutions and their derivatives to be uniformly bounded.

Keywords and Phrases: Non-linear differential equation; third order; multiple deviating arguments; bounded solutions.

---

RESUMEN

En este artículo se estudia un tipo de ecuaciones diferenciales no lineales de tercer orden con argumentos de desviación múltiple. Se establecen algunas condiciones suficientes para que todas las soluciones y sus derivadas sean uniformemente acotadas.

2010 AMS Mathematics Subject Classification: 34C25; 34K13; 34K25.

---

 

References

[1] Afuwape, A.U., Omari, P., and Zanolin, F. Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary problems. J. Math. Anal. Appl. 143, 35-56 (1989).

[2] Andres, J. Periodic boundary value problem for certain nonlinear differential equation of the third order. Math. Slovaca 35, 305-309 (1985).

[3] Fridedrichs, K. O. On nonlinear vibrations of the third-order, in: Studies in Nonlinear Vibrations Theory, Inst. Math. Mech., New York University (1949).

[4] Rauch, L.L. Oscillations of a third order nonlinear autonomous system, in: Contributions to the Theory of Nonlinear Oscillations. Ann. Math. Stud. 20, 39-88 (1950).

[5] Cronin, J. Some mathematics of biological oscillations. SIAM Rev. 19, 100-137 (1977).

[6] Gopalsamy, K. Stability and Oscillations in Delay Differential Equations of Population Dynamic, Mathematics and Its Applications, vol. 74, Kluwer Academic, Dordrecht (1992).

[7] Li, B. Uniqueness and stability of a limit cycle for a third-order dynamical system arising in neuron modeling. Nonlinear Anal. 5,13-19 (1981).

[8] Murakami, S. Asymptotic behavior of solutions of some differential equations. J. Math. Anal. Appl. 109, 534-545 (1985).

[9] Ademola, T. A., Ogundiran, M. O., Arawomo, P. O., and Adesina, O. A. Boundedness results for a certain third order nonlinear differential equation. Appl. Math. Comput. 216(10), 3044- 3049 (2010).

[10] Tunç , C. and Ergören, H. On boundedness of a certain non-linear differential equation of third order. J. Comput. Anal. Appl. 12(3), 687-94 (2010).

[11] Tunç , C. Boundedness of solutions to third-order nonlinear differential equations with bounded delay. J. Franklin Inst. 347(2), 415-425 (2010).

[12] Tunç , C. On the stability and boundedness of solutions of nonlinear vector differential equations of third order. Nonlinear Anal. 70(6), 2232-2236 (2009).

[13] Tunç , C. On the boundedness of solutions of third-order delay differential equations. Differ. Equ. (Differ. Uravn.) 44(4), 464-472 (2008).

[14] Afuwape, A.U. and Castellanos, J. E. Asymptotic and exponential stability of certain thirdorder non-linear delayed differential equations: frequency domain method. Appl. Math. Comput. 216(3), 940-950 (2010).

[15] Gao, H. and Liu, B. Almost periodic solutions for a class of Li´enard-type systems with multiple varying time delays. Appl. Math. Model. 34, 72-79 (2010).

[16] Yu, Y. and Zhao, C. Boundedness of solutions for a Lienard equation with multiple deviating arguments. Elec. J. Differential Equations. 2009(14), 1-5 (2009).

[17] Burton, T.A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orland, FL (1985).

[18] Hale, J.K. Theory of Functional Differential Equations, Springer-Verlag, New York (1977).

[19] Kuang, Y. Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York (1993).

---

Received: April 2011. Revised: February 2012.