Stability and boundedness in differential systems of third order with variable delay
DOI:
https://doi.org/10.4067/S0716-09172016000300008Keywords:
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 non-linear 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
[1] R. Reissig, G. Sansone, R. Conti, Non-linear differential equations of higher order. Translated from the German. Noordhoff International Publishing, Leyden, 1974.
[2] A. T. Ademola, P. O. Arawomo, Stability and ultimate boundedness of solutions to certain third-order differential equations. Appl. Math. E-Notes 10, pp. 61-69, (2010).
[3] A. T. Ademola, P. O. Arawomo, Stability and uniform ultimate boundedness of solutions of some third-order differential equations. Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 27, No. 1, pp. 51-59, (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. 197-211, (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. 157-166, (2013).
[6] A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, O. A. Adesina, Stability, boundedness, and existence of periodic solutions to certain third-order 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 non-linear delayed differential equations: frequency domain method. Appl. Math. Comput. 216, No. 3, pp. 940-950, (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. 329-342, (2010).
[9] 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).
[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. 95-104, (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. 64-69, (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. 143-151, (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. 82-94,? (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. 313-324, (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. 3747-3752, (2015).
[16] E. Korkmaz, C. Tunc, Convergence to non-autonomous differential equations of second order. J. Egyptian Math. Soc. 23, No. 1, pp. 27-30, (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. 1-11, (2016).
[18] B. S. Ogundare, On boundedness and stability of solutions of certain third order delay differential equation. J. Nigerian Math. Soc. 31, pp. 55-68, (2012).
[19] B. S. Ogundare, J. A. Ayanjinmi, O. A. Adesina, Bounded and L 2-solutions of certain third order non-linear differential equation with a square integrable forcing term. Kragujevac J. Math. 29, pp. 151-156, (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. 69-80, (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. 109-119, (2015).
[22] C. Qian, On global stability of third-order nonlinear differential equations. Nonlinear Anal. 42, No. 4, Ser. A: Theory Methods, pp. 651-661, (2000).
[23] M. Remili, L. D. Oudjedi, Stability and boundedness of the solutions of non-autonomous third order differential equations with delay. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 53, No. 2, pp. 139-147, (2014).
[24] A. I. Sadek, Stability and boundedness of a kind of third-order delay differential system. Appl. Math. Lett. 16, No. 5, 657-662, (2003).
[25] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations. SIAM J. Appl. Math. 19, pp. 96-102, (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. 9-19, (2000).
[27] C. Tunc, On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order. Chinese translation in Appl. Math. Mech. 20, No. 2, pp. 153-160, (1999). Appl. Math. Mech. (English Ed.) 20, No. 2, pp. 163-170, (1999).
[28] 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).
[29] C. Tunc, Boundedness of solutions of a third-order 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 third-order nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1, pp. 29-35, (2005).
[31] C. Tunc, New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations. Arab. J. Sci. Eng. Sect. A Sci. 31, No. 2, pp. 185-196, (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. 291-300, (2006).
[33] 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).
[34] 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).
[35] C. Tunc, Bounded solutions to nonlinear delay differential equations of third order. Topol. Methods Nonlinear Anal. 34, No. 1, pp. 131-139, (2009).
[36] C. Tunc, On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24, No. 3, pp.? 1-10, (2010).
[37] C. Tunc, Stability and bounded of solutions to non-autonomous delay differential equations of third order. Nonlinear Dynam. 62, No. 4, pp. 945-953, (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 non-autonomous differential equations with constant delay. Appl. Math. Inf. Sci. 7, No. 1, pp. 355-361, (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. 381-390, (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. 643-655, (2014).
[42] C. Tunc, On the stability and boundedness of certain third order nonautonomous differential equations of retarded type. Proyecciones 34, No. 2, pp. 147-159, (2015).
[43] C. Tunc, Global stability and boundedness of solutions to differential equations of third order with multiple delays. Dynam. Systems Appl. 24, pp. 467-478, (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. 109-123, (2016).
[45] C. Tunc, M. Ate, Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dynam. 45, No. 3-4, pp. 273-281, (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. 325-347, (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. 159-172, (2006).
[49] 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).
[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. 249-259, (1992).
[51] A. U. Afuwape, P. Omari, F. Zanolin, Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. J. Math. Anal. Appl. 143, No. 1, pp. 35-56, (1989).
[52] J. Andres, Periodic boundary value problem for certain nonlinear differential equations of the third order. Math. Slovaca 35, No. 3, 305-309, (1985).
[53] 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).
[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. 575-586, Hadronic Press, Palm Harbor, FL, (1999).
[55] J. O. C. Ezeilo, J. Onyia, Nonresonant oscillations for some third-order differential equations. J. Nigerian Math. Soc. 3 (1984), 83-96 (1986).
[56] K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28 (2006), no. 3, 739-746.
[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), 7151-7164.
[58] S. J. Linz, On hyperjerky systems. Chaos Solitons Fractals 37 (2008), no. 3, 741-747.
[59] J. Cronin-Scanlon, 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. 39-88. 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. 452-482, (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 third-order differential equations. Appl. Math. E-Notes 10, pp. 61-69, (2010).
[3] A. T. Ademola, P. O. Arawomo, Stability and uniform ultimate boundedness of solutions of some third-order differential equations. Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 27, No. 1, pp. 51-59, (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. 197-211, (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. 157-166, (2013).
[6] A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, O. A. Adesina, Stability, boundedness, and existence of periodic solutions to certain third-order 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 non-linear delayed differential equations: frequency domain method. Appl. Math. Comput. 216, No. 3, pp. 940-950, (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. 329-342, (2010).
[9] 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).
[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. 95-104, (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. 64-69, (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. 143-151, (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. 82-94,? (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. 313-324, (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. 3747-3752, (2015).
[16] E. Korkmaz, C. Tunc, Convergence to non-autonomous differential equations of second order. J. Egyptian Math. Soc. 23, No. 1, pp. 27-30, (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. 1-11, (2016).
[18] B. S. Ogundare, On boundedness and stability of solutions of certain third order delay differential equation. J. Nigerian Math. Soc. 31, pp. 55-68, (2012).
[19] B. S. Ogundare, J. A. Ayanjinmi, O. A. Adesina, Bounded and L 2-solutions of certain third order non-linear differential equation with a square integrable forcing term. Kragujevac J. Math. 29, pp. 151-156, (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. 69-80, (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. 109-119, (2015).
[22] C. Qian, On global stability of third-order nonlinear differential equations. Nonlinear Anal. 42, No. 4, Ser. A: Theory Methods, pp. 651-661, (2000).
[23] M. Remili, L. D. Oudjedi, Stability and boundedness of the solutions of non-autonomous third order differential equations with delay. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 53, No. 2, pp. 139-147, (2014).
[24] A. I. Sadek, Stability and boundedness of a kind of third-order delay differential system. Appl. Math. Lett. 16, No. 5, 657-662, (2003).
[25] K. E. Swick, Asymptotic behavior of the solutions of certain third order differential equations. SIAM J. Appl. Math. 19, pp. 96-102, (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. 9-19, (2000).
[27] C. Tunc, On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order. Chinese translation in Appl. Math. Mech. 20, No. 2, pp. 153-160, (1999). Appl. Math. Mech. (English Ed.) 20, No. 2, pp. 163-170, (1999).
[28] 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).
[29] C. Tunc, Boundedness of solutions of a third-order 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 third-order nonlinear differential equations. J. Appl. Math. Stoch. Anal., No. 1, pp. 29-35, (2005).
[31] C. Tunc, New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations. Arab. J. Sci. Eng. Sect. A Sci. 31, No. 2, pp. 185-196, (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. 291-300, (2006).
[33] 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).
[34] 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).
[35] C. Tunc, Bounded solutions to nonlinear delay differential equations of third order. Topol. Methods Nonlinear Anal. 34, No. 1, pp. 131-139, (2009).
[36] C. Tunc, On the stability and boundedness of solutions of nonlinear third order differential equations with delay. Filomat 24, No. 3, pp.? 1-10, (2010).
[37] C. Tunc, Stability and bounded of solutions to non-autonomous delay differential equations of third order. Nonlinear Dynam. 62, No. 4, pp. 945-953, (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 non-autonomous differential equations with constant delay. Appl. Math. Inf. Sci. 7, No. 1, pp. 355-361, (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. 381-390, (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. 643-655, (2014).
[42] C. Tunc, On the stability and boundedness of certain third order nonautonomous differential equations of retarded type. Proyecciones 34, No. 2, pp. 147-159, (2015).
[43] C. Tunc, Global stability and boundedness of solutions to differential equations of third order with multiple delays. Dynam. Systems Appl. 24, pp. 467-478, (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. 109-123, (2016).
[45] C. Tunc, M. Ate, Stability and boundedness results for solutions of certain third order nonlinear vector differential equations. Nonlinear Dynam. 45, No. 3-4, pp. 273-281, (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. 325-347, (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. 159-172, (2006).
[49] 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).
[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. 249-259, (1992).
[51] A. U. Afuwape, P. Omari, F. Zanolin, Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems. J. Math. Anal. Appl. 143, No. 1, pp. 35-56, (1989).
[52] J. Andres, Periodic boundary value problem for certain nonlinear differential equations of the third order. Math. Slovaca 35, No. 3, 305-309, (1985).
[53] 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).
[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. 575-586, Hadronic Press, Palm Harbor, FL, (1999).
[55] J. O. C. Ezeilo, J. Onyia, Nonresonant oscillations for some third-order differential equations. J. Nigerian Math. Soc. 3 (1984), 83-96 (1986).
[56] K. E. Chlouverakis, J. C. Sprott, Chaotic hyperjerk systems. Chaos Solitons Fractals 28 (2006), no. 3, 739-746.
[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), 7151-7164.
[58] S. J. Linz, On hyperjerky systems. Chaos Solitons Fractals 37 (2008), no. 3, 741-747.
[59] J. Cronin-Scanlon, 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. 39-88. 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. 452-482, (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
2017-03-23
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. 317-338, Mar. 2017.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
