A NEW BOUNDEDNESS RESULT TO NONLINEAR DIFFERENTIAL EQUATIONS OF THIRD ORDER WITH FINITE LAG

CEMIL TUNC

Title	A NEW BOUNDEDNESS RESULT TO NONLINEAR DIFFERENTIAL EQUATIONS OF THIRD ORDER WITH FINITE LAG
Publication Type	Journal Article
Year of Publication	2009
Authors	TUNC, CEMIL
Secondary Title	Communications in Applied Analysis
Volume	13
Issue	1
Start Page	1
Pagination	10
Date Published	01/2009
Type of Work	scientific: mathematics
ISSN	1083–2564
AMS	34K20.
Abstract	Criteria for boundedness of solutions to the nonlinear third order delay differential equation are obtained by Lyapunov’s second method. By introducing a Lyapunov functional, sufficient conditions are established that guarantee that all solutions of this equation are bounded. An example is also given to illustrate the importance of result obtained. Our findings improve a result existing in the literature to boundedness of solutions for this delay differential equation.
URL	http://www.acadsol.eu/en/articles/13/1/1.pdf
Short Title	A NEW BOUNDEDNESS RESULT
Refereed Designation	Refereed
Full Text	REFERENCES [1] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations. Mathematics in Science and Engineering, 178. Academic Press, Inc., Orlando, FL, 1985. [2] L. E. Elsgolts, Introduction to the Theory of Differential Equations with Deviating Arguments. Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. [3] L. E. Elsgolts S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Translated from the Russian by John L. Casti. Mathematics in Science and Engineering, Vol. 105. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. [4] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht, 1992. [5] M. Gregus, Third Order Linear Differential Equations. Translated from the Slovak by J.Dravecky. Mathematics and its Applications (East European Series), 22. D. Reidel Publishing Co., Dordrecht, 1987. [6] J. Hale, Theory of Functional Differential Equations. Springer-Verlag, New York-Heidelberg, 1977. [7] J. Hale, S. M. Verduyn Lunel, Introduction to Functional-Differential Equations. Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. [8] V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 1999. [9] V. B. Kolmanovskii, V. R. Nosov, Stability of Functional-Differential Equations. Mathematics in Science and Engineering, 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. [10] N. N. Krasovskii, Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Translated by J. L. Brenner Stanford University Press, Stanford, Calif. 1963. [11] A. M. Lyapunov, Stability of Motion. Mathematics in Science and Engineering, Vol. 30 Academic Press, New York-London, 1966. [12] O. Palusinski, P. Stern, E. Wall , M. Moe, Comments on “An energy metric algorithm for the generation of Liapuno functions”. IEEE Transactions on Automatic Control, Volume 14, Issue 1, (1969), 110 -111. [13] R. Reissig, G. Sansone, R. Conti, Non-linear Differential Equations of Higher Order, Translated from the German. Noordhoff International Publishing, Leyden, 1974. [14] A. I. Sadek, Stability and boundedness of a kind of third-order delay differential system. Appl. Math. Lett. 16 (2003), no. 5, 657-662. [15] C. Tun ̧c, A new boundedness theorem for a class of second order differential equations. Arab. J. Sci. Eng. Sect. A Sci. 33 (2008), no. 1, 83–92. [16] C. Tun ̧c, Stability and boundedness of solutions of nonlinear differential equations of thirdorder with delay. Differ. Uravn. Protsessy Upr. 2007, no. 3, 1–13. [17] C. Tun ̧c, On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument. Commun. Appl. Anal. 11 (2007), no. 3–4, 515–527. [18] T. Yoshizawa, Stability Theory by Liapunov’s Second Method. The Mathematical Society of Japan, Tokyo, 1966.