Stability and boundedness in differential systems of third order with variable delay
DOI:
https://doi.org/10.4067/S071609172016000300008Keywords:
Globally asymptotic stability, boundedness, Lyapunov functional, delay, differential system, third order, estabilidad globalmente asintótica, acotamiento, funcional de Lyapunov, retardo, sistema diferencial, tercer orden.Abstract
In this paper, we consider a nonlinear system of differential equations ofthird order with variable delay. We discuss the globally asymptotic stability/uniformly stability, boundedness and uniformly boundedness ofsolutionsfor the considered system. The technique ofproofs involves defining an appropriate Lyapunov functional. The obtained results include and improve the results in literature.References
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[27] C. Tunc, On the boundedness and periodicity of the solutions of a certain vector differential equation of thirdorder. Chinese translation in Appl. Math. Mech. 20, No. 2, pp. 153160, (1999). Appl. Math. Mech. (English Ed.) 20, No. 2, pp. 163170, (1999).
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[29] C. Tunc, Boundedness of solutions of a thirdorder nonlinear differential equation. JIPAM. J. Inequal. Pure Appl. Math. 6, o. 1, Article 3, 6 pp., (2005).
[30] C. Tunc, On the asymptotic behavior of solutions of certain thirdorder nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1, pp. 2935, (2005).
[31] C. Tunc, New results about stability and boundedness of solutions of certain nonlinear thirdorder delay differential equations. Arab. J. Sci. Eng. Sect. A Sci. 31, No. 2, pp. 185196, (2006).
[32] C. Tunc, On the boundedness of solutions of certain nonlinear vector differential equations of third order. Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 49 (97), No. 3, pp. 291300, (2006).
[33] C. Tunc, On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dynam. 57, No. 12, pp. 97106, (2009).
[34] C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order. Nonlinear Anal. 70, No. 6, pp. 22322236, (2009).
[35] C. Tunc, Bounded solutions to nonlinear delay differential equations of third order. Topol. Methods Nonlinear Anal. 34, No. 1, pp. 131139, (2009).
[36] C. Tunc, On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24, No. 3, pp.? 110, (2010).
[37] C. Tunc, Stability and bounded of solutions to nonautonomous delay differential equations of third order. Nonlinear Dynam. 62, No. 4, pp. 945953, (2010).
[38] 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).
[39] C. Tunc, Stability and boundedness for a kind of nonautonomous differential equations with constant delay. Appl. Math. Inf. Sci. 7, No. 1, pp. 355361, (2013).
[40] C. Tunc, Stability and boundedness of the nonlinear differential equations of third order with multiple deviating arguments. Afr. Mat. 24, No. 3, pp. 381390, (2013).
[41] 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. 643655, (2014).
[42] C. Tunc, On the stability and boundedness of certain third order nonautonomous differential equations of retarded type. Proyecciones 34, No. 2, pp. 147159, (2015).
[43] C. Tunc, Global stability and boundedness of solutions to differential equations of third order with multiple delays. Dynam. Systems Appl. 24, pp. 467478, (2015).
[44] C. Tunc, Boundedness of solutions to certain system of differential equations with multiple delays. Mathematical Modeling and Applications in Nonlinear Dynamics. Springer Book Series, Chapter 5, pp. 109123, (2016).
[45] C. Tunc, M. Ate, Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dynam. 45, No. 34, pp. 273281, (2006).
[46] C. Tunc, M. Gzen, Stability and uniform boundedness in multidelay functional differential equations of third order. Abstr. Appl. Anal., Art. ID 248717, 7 pp. –, (2013).
[47] C. Tunc, S. A. Mohammed, On the qualitative properties of differential equations of third order with retarded argument. Proyecciones 33, No.? 3, pp. 325347, (2014).
[48] C. Tunc; E. Tunc, New ultimate boundedness and periodicity results for certain third order nonlinear vector differential equations. Math. J. Okayama Univ. 48, pp. 159172, (2006).
[49] L. Zhang; L. Yu, Global asymptotic stability of certain thirdorder nonlinear differential equations. Math. Methods Appl. Sci. Math. Methods Appl. Sci. 36, No. 14, pp. 18451850, (2013).
[50] Y. F. Zhu, On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differential Equations 8, No. 2, pp. 249259, (1992).
[51] 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. 3556, (1989).
[52] J. Andres, Periodic boundary value problem for certain nonlinear differential equations of the third order. Math. Slovaca 35, No. 3, 305309, (1985).
[53] K. O. Fridedrichs, On nonlinear vibrations of third order. Studies in Nonlinear Vibration Theory, pp. 65103. Institute for Mathematics and Mechanics, New York University, (1946).
[54] A. O. E. Animalu, J. O. C. Ezeilo, Some third order differential equations in physics. Fundamental open problems in science at the end of the millennium, Vol. IIII (Beijing, 1997), pp. 575586, Hadronic Press, Palm Harbor, FL, (1999).
[55] J. O. C. Ezeilo, J. Onyia, Nonresonant oscillations for some thirdorder differential equations. J. Nigerian Math. Soc. 3 (1984), 8396 (1986).
[56] K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28 (2006), no. 3, 739746.
[57] R. Eichhorn, S. J. Linz, P. Hnggi, Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows. Phys Rev E 58 (1998), 71517164.
[58] S. J. Linz, On hyperjerky systems. Chaos Solitons Fractals 37 (2008), no. 3, 741747.
[59] J. CroninScanlon, Some mathematics of biological oscillations. SIAM Rev. 19 (1977), no. 1, 100138.
[60] L. L. Rauch, Oscillation of a third order nonlinear autonomous system. Contributions to the Theory of Nonlinear Oscillations, pp. 3988. Annals of Mathematics Studies, no. 20. Princeton University Press, Princeton, N. J., (1950).
[61] H. Smith, An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics, 57. Springer, New York, (2011).
[62] J. Hale, Sufficient conditions for stability and instability of autonomous functional differential equations. J. Differential Equations 1, pp. 452482, (1965)
[63] T. Yoshizawa, Stability theory by Liapunov’s second method. Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo (1966).
[64] R. Bellman, Richard Introduction to matrix analysis. Reprint of the second edition 1970. With a foreword by Gene Golub. Classics in Applied Mathematics, 19. Society for Industrial and Applied Mathematics? (SIAM), Philadelphia, PA, (1997).
[2] A. T. Ademola, P. O. Arawomo, Stability and ultimate boundedness of solutions to certain thirdorder differential equations. Appl. Math. ENotes 10, pp. 6169, (2010).
[3] A. T. Ademola, P. O. Arawomo, Stability and uniform ultimate boundedness of solutions of some thirdorder differential equations. Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 27, No. 1, pp. 5159, (2011).
[4] A. T. Ademola, P. O. Arawomo, Asymptotic behaviour of solutions of third order nonlinear differential equations. Acta Univ. Sapientiae Math. 3, No. 2, pp. 197211, (2011).
[5] A. T. Ademola, P. O. Arawomo, Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order. Math. J. Okayama Univ. 55, pp. 157166, (2013).
[6] A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, O. A. Adesina, Stability, boundedness, and existence of periodic solutions to certain thirdorder delay differential equations with multiple deviating arguments. Int. J. Differ. Equ., Art. ID 213935, 12 pp...., (2015).
[7] A. U. Afuwape, J. E. Castellanos, Asymptotic and exponential stability of certain third order nonlinear delayed differential equations: frequency domain method. Appl. Math. Comput. 216, No. 3, pp. 940950, (2010).
[8] A. U. Afuwape, M. O. Omeike, Stability and boundedness of solutions of a kind of third order delay differential equations. Comput. Appl. Math. 29, No. 3, pp. 329342, (2010).
[9] S. Ahmad, M. Rama Mohana Rao, Theory of ordinary differential equations. With applications in biology and engineering. Affiliated EastWest Press Pvt. Ltd., New Delhi, (1999).
[10] Y. Bai, C. Guo, New results on stability and boundedness of third order nonlinear delay differential equations. Dynam. Systems Appl. 22, No. 1, pp. 95104, (2013).
[11] 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. 6469, (1960).
[12] J. O. C. Ezeilo, H. O. Tjumla, Further results for a system of third order differential equations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58, No. 2, pp. 143151, (1975).
[13] J. R. Graef, D. Beldjerd, M. Remili, On stability, ultimate boundedness, and existence of periodic solutions of certain third order differential equations with delay. PanAmer. Math. J. 25, No. 1, pp. 8294,? (2015).
[14] J. R. Graef, L. D. Oudjedi, M. Remili, Stability and square integrability of solutions of nonlinear third order differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 22, No. 4, pp. 313324, (2015).
[15] ] J. R. Graef, C. Tunc, Global asymptotic stability and boundedness of certain multidelay functional differential equations of third order. Math. Methods Appl. Sci. 38, No. 17, pp. 37473752, (2015).
[16] E. Korkmaz, C. Tunc, Convergence to nonautonomous differential equations of second order. J. Egyptian Math. Soc. 23, No. 1, pp. 2730, (2015).
[17] A. M. Mahmoud, C. Tunc, Stability and boundedness of solutions of a certain n dimensional nonlinear delay differential system of thirdorder. Adv. Pure Appl. Math. 7(1), pp. 111, (2016).
[18] B. S. Ogundare, On boundedness and stability of solutions of certain third order delay differential equation. J. Nigerian Math. Soc. 31, pp. 5568, (2012).
[19] B. S. Ogundare, J. A. Ayanjinmi, O. A. Adesina, Bounded and L 2solutions of certain third order nonlinear differential equation with a square integrable forcing term. Kragujevac J. Math. 29, pp. 151156, (2006)
[20] A. L. Olutimo, Stability and ultimate boundedness of solutions of a certain third order nonlinear vector differential equation. J. Nigerian Math. Soc. 31, pp. 6980, (2012).
[21] M. O. Omeike, Stability and boundedness of solutions of a certain system of third order nonlinear delay differential equations. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 54, No. 1, pp. 109119, (2015).
[22] C. Qian, On global stability of thirdorder nonlinear differential equations. Nonlinear Anal. 42, No. 4, Ser. A: Theory Methods, pp. 651661, (2000).
[23] M. Remili, L. D. Oudjedi, Stability and boundedness of the solutions of nonautonomous third order differential equations with delay. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 53, No. 2, pp. 139147, (2014).
[24] A. I. Sadek, Stability and boundedness of a kind of thirdorder delay differential system. Appl. Math. Lett. 16, No. 5, 657662, (2003).
[25] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations. SIAM J. Appl. Math. 19, pp. 96102, (1970).
[26] H. O. Tejumola, B. Tchegnani, Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations. J. Nigerian Math. Soc. 19, pp. 919, (2000).
[27] C. Tunc, On the boundedness and periodicity of the solutions of a certain vector differential equation of thirdorder. Chinese translation in Appl. Math. Mech. 20, No. 2, pp. 153160, (1999). Appl. Math. Mech. (English Ed.) 20, No. 2, pp. 163170, (1999).
[28] C. Tunc, Uniform ultimate boundedness of the solutions of thirdorder nonlinear differential equations. Kuwait J. Sci. Engrg. 32, No. 1, pp. 3948, (2005).
[29] C. Tunc, Boundedness of solutions of a thirdorder nonlinear differential equation. JIPAM. J. Inequal. Pure Appl. Math. 6, o. 1, Article 3, 6 pp., (2005).
[30] C. Tunc, On the asymptotic behavior of solutions of certain thirdorder nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1, pp. 2935, (2005).
[31] C. Tunc, New results about stability and boundedness of solutions of certain nonlinear thirdorder delay differential equations. Arab. J. Sci. Eng. Sect. A Sci. 31, No. 2, pp. 185196, (2006).
[32] C. Tunc, On the boundedness of solutions of certain nonlinear vector differential equations of third order. Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 49 (97), No. 3, pp. 291300, (2006).
[33] C. Tunc, On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument. Nonlinear Dynam. 57, No. 12, pp. 97106, (2009).
[34] C. Tunc, On the stability and boundedness of solutions of nonlinear vector differential equations of third order. Nonlinear Anal. 70, No. 6, pp. 22322236, (2009).
[35] C. Tunc, Bounded solutions to nonlinear delay differential equations of third order. Topol. Methods Nonlinear Anal. 34, No. 1, pp. 131139, (2009).
[36] C. Tunc, On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24, No. 3, pp.? 110, (2010).
[37] C. Tunc, Stability and bounded of solutions to nonautonomous delay differential equations of third order. Nonlinear Dynam. 62, No. 4, pp. 945953, (2010).
[38] 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).
[39] C. Tunc, Stability and boundedness for a kind of nonautonomous differential equations with constant delay. Appl. Math. Inf. Sci. 7, No. 1, pp. 355361, (2013).
[40] C. Tunc, Stability and boundedness of the nonlinear differential equations of third order with multiple deviating arguments. Afr. Mat. 24, No. 3, pp. 381390, (2013).
[41] 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. 643655, (2014).
[42] C. Tunc, On the stability and boundedness of certain third order nonautonomous differential equations of retarded type. Proyecciones 34, No. 2, pp. 147159, (2015).
[43] C. Tunc, Global stability and boundedness of solutions to differential equations of third order with multiple delays. Dynam. Systems Appl. 24, pp. 467478, (2015).
[44] C. Tunc, Boundedness of solutions to certain system of differential equations with multiple delays. Mathematical Modeling and Applications in Nonlinear Dynamics. Springer Book Series, Chapter 5, pp. 109123, (2016).
[45] C. Tunc, M. Ate, Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dynam. 45, No. 34, pp. 273281, (2006).
[46] C. Tunc, M. Gzen, Stability and uniform boundedness in multidelay functional differential equations of third order. Abstr. Appl. Anal., Art. ID 248717, 7 pp. –, (2013).
[47] C. Tunc, S. A. Mohammed, On the qualitative properties of differential equations of third order with retarded argument. Proyecciones 33, No.? 3, pp. 325347, (2014).
[48] C. Tunc; E. Tunc, New ultimate boundedness and periodicity results for certain third order nonlinear vector differential equations. Math. J. Okayama Univ. 48, pp. 159172, (2006).
[49] L. Zhang; L. Yu, Global asymptotic stability of certain thirdorder nonlinear differential equations. Math. Methods Appl. Sci. Math. Methods Appl. Sci. 36, No. 14, pp. 18451850, (2013).
[50] Y. F. Zhu, On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system. Ann. Differential Equations 8, No. 2, pp. 249259, (1992).
[51] 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. 3556, (1989).
[52] J. Andres, Periodic boundary value problem for certain nonlinear differential equations of the third order. Math. Slovaca 35, No. 3, 305309, (1985).
[53] K. O. Fridedrichs, On nonlinear vibrations of third order. Studies in Nonlinear Vibration Theory, pp. 65103. Institute for Mathematics and Mechanics, New York University, (1946).
[54] A. O. E. Animalu, J. O. C. Ezeilo, Some third order differential equations in physics. Fundamental open problems in science at the end of the millennium, Vol. IIII (Beijing, 1997), pp. 575586, Hadronic Press, Palm Harbor, FL, (1999).
[55] J. O. C. Ezeilo, J. Onyia, Nonresonant oscillations for some thirdorder differential equations. J. Nigerian Math. Soc. 3 (1984), 8396 (1986).
[56] K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28 (2006), no. 3, 739746.
[57] R. Eichhorn, S. J. Linz, P. Hnggi, Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows. Phys Rev E 58 (1998), 71517164.
[58] S. J. Linz, On hyperjerky systems. Chaos Solitons Fractals 37 (2008), no. 3, 741747.
[59] J. CroninScanlon, Some mathematics of biological oscillations. SIAM Rev. 19 (1977), no. 1, 100138.
[60] L. L. Rauch, Oscillation of a third order nonlinear autonomous system. Contributions to the Theory of Nonlinear Oscillations, pp. 3988. Annals of Mathematics Studies, no. 20. Princeton University Press, Princeton, N. J., (1950).
[61] H. Smith, An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics, 57. Springer, New York, (2011).
[62] J. Hale, Sufficient conditions for stability and instability of autonomous functional differential equations. J. Differential Equations 1, pp. 452482, (1965)
[63] T. Yoshizawa, Stability theory by Liapunov’s second method. Publications of the Mathematical Society of Japan, No. 9. The Mathematical Society of Japan, Tokyo (1966).
[64] R. Bellman, Richard Introduction to matrix analysis. Reprint of the second edition 1970. With a foreword by Gene Golub. Classics in Applied Mathematics, 19. Society for Industrial and Applied Mathematics? (SIAM), Philadelphia, PA, (1997).
Published
20170323
How to Cite
[1]
C. Tunc, “Stability and boundedness in differential systems of third order with variable delay”, Proyecciones (Antofagasta, On line), vol. 35, no. 3, pp. 317338, Mar. 2017.
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