# Stability and boundedness in differential systems of third order with variable delay

Palabras clave:
Globally asymptotic stability, boundedness, Lyapunov functional, delay, differential system, third order, estabilidad globalmente asintótica, acotamiento, funcional de Lyapunov, retardo, sistema diferencial, tercer orden.

### Resumen

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.### Citas

[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).

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2017-03-23

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*35*(3), 317-338. https://doi.org/10.4067/S0716-09172016000300008

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