On the qualitative properties of differential equations of third order with retarded argument

Cemil Tunç, Sizar Abid Mohammed


By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument.

Palabras clave

Stability; boundedness; Liapunov-Krasovskii functional approach; retarded argument; estabilidad; condición de borde; enfoque funcional de Liapunov-Krasovskii; argumento con retardo.

Texto completo:



T. A. Ademola; P. O. Arawomo, Asymptotic behaviour of solutions‏ of third order nonlinear differential equations. Acta Univ. Sapientiae‏ Math. 3, No. 2, pp. 197-211, (2011).‏

A. U. Afuwape; J. E. Castellanos, Asymptotic and exponential stability of certain third-order non-linear delayed differential equations:‏ frequency domain method. Appl. Math. Comput. 216, No. 3, pp. 940-950, (2010).‏

A. U. Afuwape; P. Omari; F. Zanolin, Nonlinear perturbations of differential operators with nontrivial kernel and applications to thirdorder periodic boundary value problems. J. Math. Anal. Appl. 143,‏ No. 1, pp. 35-56, (1989).‏

S. Ahmad; M. Rama Mohana Rao, Theory of ordinary differential‏ equations. With applications in biology and engineering. Affiliated‏ East-West Press Pvt. Ltd., New Delhi, (1999).‏

T. A. Burton, Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando, (1985).‏

K. E. Chlouverakis; J. C. Sprott, Chaotic hyperjerk systems. Chaos‏ Solitons Fractals 28, No. 3, pp. 739-746, (2006).‏

E. N. Chukwu, On boundedness of solutions of third order differential‏ equations. Ann. Math. Pura. Appl. 104, pp. 123-149, (1975).‏

J. Cronin-Scanlon, Some mathematics of biological oscillations. SIAM‏ Rev. 19, No. 1, pp. 100-138, (1977).‏

Z. Elhadj; J. C. Sprott, Boundedness of certain forms of jerky dynamics. Qual. Theory Dyn. Syst. 11, No. 2, pp. 199-213, (2012).‏

R. Eichhorn; S. J. Linz; P. Hanggi, Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic‏ flows. Phys Rev E 58, pp. 7151-7164, (1998).

Elsgolts, L. E.; Norkin, S. B., 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, New York-London, (1973).‏

J. O. C. Ezeilo, On the stability of solutions of certain differential‏ equations of the third order. Quart. J. Math. Oxford Ser. (2) 11, pp.

-69, (1960).

Ezeilo, J. O. C., A generalization of a boundedness theorem for a‏ certain third-order differential equation. Proc. Cambridge Philos. Soc.‏ 63, pp. 735-742, (1967).‏

K. O. Fridedrichs, On nonlinear vibrations of third order. Studies in‏ Nonlinear Vibration Theory, pp. 65-103. Institute for Mathematics and‏ Mechanics, New York University, (1946).‏

J. Hale, Sufficient conditions for stability and instability of autonomous‏ functional-differential equations. J. Differential Equations 1, pp. 452-482, (1965).‏

T. Hara, On the uniform ultimate boundedness of the solutions of‏ certain third order differential equations. J. Math. Anal. Appl. 80, No.‏ 2, pp. 533-544, (1981).‏

N. N. Krasovski´ i, Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Stanford, Calif.: Stanford University Press (1963).‏

A. M. Liapunov, Stability of motion. With a contribution by V. A. Pliss‏ and an introduction by V. P. Basov. Translated from the Russian by‏ Flavian Abramovici and Michael Shimshoni. Mathematics in Science‏ and Engineering, Vol. 30 Academic Press, New York-London (1966).‏

S. J. Linz, On hyperjerky systems. Chaos Solitons Fractals 37, No. 3,‏ pp. 741-747, (2008).‏

B. Mehri; D. Shadman, Boundedness of solutions of certain third order differential equation. Math. Inequal. Appl. 2, No.4, pp. 545-549,‏ (1999).‏

F. W. Meng, Ultimate boundedness results for a certain system of‏ third order nonlinear differential equations. J. Math. Anal. Appl. 177,‏ No. 2, pp. 496-509, (1993).

M. O.Omeike; A. U. Afuwape, New result on the ultimate boundedness‏ of solutions of certain thir d-order vector differential equations. Acta‏ Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 49, No. 1, pp. 55-61,‏ (2010).‏

C. Qian, On global stability of third-order nonlinear differential equations. Nonlinear Anal. 42, No. 4, Ser. A: Theory Methods, pp. 651-661,‏ (2000).‏

L. L. Rauch, Oscillation of a third order nonlinear autonomous system. Contributions to the Theory of Nonlinear Oscillations, pp. 39-88.‏ Annals of Mathematics Studies, no. 20. Princeton University Press,‏ Princeton, N.J., (1950).‏

R. Reissig; G. Sansone; R. Conti, Non-linear differential equations of‏ higher order. Translated from the German. Noordhoff International‏ Publishing, Leyden, (1974).‏

H. Smith, An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics, 57. Springer,‏ New York, (2011).‏

K. E. Swick, Asymptotic behavior of the solutions of certain third order‏ differential equations. SIAM J. Appl. Math. 19, pp. 96-102, (1970).‏

H. O. Tejumola, A note on the boundedness and the stability of solutions of certain third-order differential equations. Ann. Mat. Pura‏ Appl. (4) 92, pp. 65-75, (1972).‏

C. Tunc, Global stability of solutions of certain third-order nonlinear‏ differential equations. Panamer. Math. J. 14, No. 4, pp. 31-35, (2004).‏

C. Tunc, Uniform ultimate boundedness of the solutions of third-order‏ nonlinear differential equations. Kuwait J. Sci. Engrg. 32, No. 1, pp.‏ 39-48, (2005).‏

C. Tunc, Boundedness of solutions of a third-order nonlinear differential equation. JIPAM. J. Inequal. Pure Appl. Math. 6, No. 1, Article‏ 3, 6 pp., (2005)‏

C. Tunc, On the asymptotic behavior of solutions of certain third-order‏ nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1,‏ pp. 29-35, (2005).

C. Tunc, Some instability results on certain third order nonlinear vector differential equations. Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007),‏ no. 1, 109-122.‏

C. Tunc, On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear‏ Dynam. 57, No. 1-2, pp. 97-106, (2009).‏

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

C. Tunc, Stability and bounded of solutions to non-autonomous delay‏ differential equations of third order. Nonlinear Dynam. 62 (2010), No.‏ 4, pp. 945-953, (2010).‏

C. Tunc, On some qualitative behaviors of solutions to a kind of third‏ order nonlinear delay differential equations. Electron. J. Qual. Theory‏ Differ. Equ., No. 12, 19, pp. –, (2010).‏

C. Tunc, Stability and boundedness for a kind of non-autonomous‏ differential equations with constant delay. Appl. Math. Inf. Sci. 7, No.‏ 1, pp. 355-361, (2013).‏

C. Tunc, Stability and boundedness of the nonlinear differential equations of third order with multiple deviating arguments. Afr. Mat. 24,‏ No. 3, pp. 381-390, (2013).‏

C. Tunc, On the qualitative behaviors of solutions of some differential equations of higher order with multiple deviating arguments. J.‏ Franklin Inst. 351, No. 2, pp. 643-655, (2014).‏

C. Tunc; M. Ates, Stability and boundedness results for solutions of‏ certain third order nonlinear vector differential equations. Nonlinear‏ Dynam. 45, No. 3-4, pp. 273-281, (2006).‏

C. Tunc; H. Ergoren, Uniformly boundedness of a class of non-linear‏ differential equations of third order with multiple deviating arguments.‏ Cubo 14, No. 3, 63-69, (2012).‏

E. T. Wall; M. L. Moe, An energy metric algorithm for the generation‏ of Liapunov functions. IEEE Trans. Automat. Control. 13 (1), pp. 121-122, (1968).

L. Zhang; L. Yu, Global asymptotic stability of certain third-order nonlinear differential equations. Math. Methods Appl. Sci. Math. Methods‏ Appl. Sci. 36, No. 14, pp. 1845-1850, (2013)‏

DOI: http://dx.doi.org/10.4067/S0716-09172014000300007

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