On the instability of solutions to a Liénard type equation with multiple deviating arguments. (English) Zbl 1309.34128
Summary: This paper investigates the instability of the zero solution to a modified Liénard equation with multiple constant deviating arguments. By means of the Lyapunov-Krasovskii functional approach, we obtain a new result on the topic and two examples for the illustration of the result to be obtained.
MSC:
| 34K20 | Stability theory of functional-differential equations |
Keywords:
Liénard type equation; instability; Lyapunov-Krasovskii functional; multiple deviating argumentsReferences:
| [1] | Ahmad, S., Rama Mohana Rao, M.: Theory of ordinary differential equations. With applications in biology and engineering. Affiliated East-West Press Pvt. Ltd., New Delhi (1999) · Zbl 0901.34041 |
| [2] | Burton, T.A.: Stability and periodic solutions of ordinary and functional differential equations. Academic Press, Orlando (1985) · Zbl 0635.34001 |
| [3] | Boussaada, I., Chouikha, A.R.: Existence of periodic solution for perturbed generalized Liénard equations. Electron. J. Differ. Equ. 140, 10 (2006) · Zbl 1118.34037 |
| [4] | Èl’sgol’ts, L.: Introduction to the theory of differential equations with deviating arguments. Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, London (1966) · Zbl 0133.33502 |
| [5] | Èl’sgol’ts, 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 [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1973) · Zbl 0287.34073 |
| [6] | Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht (1992) · Zbl 0752.34039 |
| [7] | Graef, John R.: Some boundedness results for nonautonomous second order nonlinear differential equations. Abh. Math. Sem. Univ. Hamburg 49, 70-73 (1979) · Zbl 0455.34022 · doi:10.1007/BF02950647 |
| [8] | Haddock, J.R., Ko, Y.: Liapunov-Razumikhin functions and an instability theorem for autonomous functional-differential equations with finite delay. Second Geoffrey J. butler memorial conference in differential equations and mathematical biology (Edmonton, AB, 1992). Rocky Mountain J. Math. 25(1), 261-267 (1995) · Zbl 0829.34068 · doi:10.1216/rmjm/1181072282 |
| [9] | Haddock, J.R., Zhao, J.: Instability for autonomous and periodic functional-differential equations with finite delay. Funkcial. Ekvac. 39(3), 553-570 (1996) · Zbl 0872.34049 |
| [10] | Haddock, J.R., Zhao, J.: Instability for functional differential equations. Math. Nachr. 279(13—-14), 1491-1504 (2006) · Zbl 1130.34050 · doi:10.1002/mana.200410434 |
| [11] | Hale, J.K.: Sufficient conditions for stability and instability of autonomous functional-differential equations. \[J\] J. Differ. Equ. 1, 452-482 (1965) · Zbl 0135.30301 · doi:10.1016/0022-0396(65)90005-7 |
| [12] | Hale, J.K.: Theory of functional differential equations. Springer-Verlag, Heidelberg (1977) · Zbl 0352.34001 · doi:10.1007/978-1-4612-9892-2 |
| [13] | Hara, T., Yoneyama, T.: On the global center of generalized Liénard equation and its application to stability problems. Funkcial. Ekvac. 28(2), 171-192 (1985) · Zbl 0585.34038 |
| [14] | Heidel, J.W.: Global asymptotic stability of a generalized Liénard equation. SIAM J. Appl. Math. 19, 629-636 (1970) · Zbl 0186.41701 · doi:10.1137/0119061 |
| [15] | Hu, G.-P., Li, W.-T., Yan, Xi-P: Hopf bifurcation and stability of periodic solutions in the delayed Liénard equation. Internat. J. Bifur. Chaos Appl. Sci. Eng. 18(10), 3147-3157 (2008) · Zbl 1165.34404 · doi:10.1142/S0218127408022317 |
| [16] | Kato, J.: On a boundedness condition for solutions of a generalized Liénard equation. J. Differ. Equ. 65(2), 269-286 (1986) · Zbl 0612.34042 · doi:10.1016/0022-0396(86)90038-0 |
| [17] | Kato, J.: A simple boundedness theorem for a Liénard equation with damping. Ann. Polon. Math. 51, 183-188 (1990) · Zbl 0721.34063 |
| [18] | Ko, Y.: The instability for functional-differential equations. J. Korean Math. Soc. 36(4), 757-771 (1999) · Zbl 0940.34060 |
| [19] | Kolmanovskii, V., Myshkis, A.: Introduction to the theory and applications of functional differential equations. Kluwer Academic Publishers, Dordrecht (1999) · Zbl 0917.34001 · doi:10.1007/978-94-017-1965-0 |
| [20] | Krasovskii, N.N.: Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay. Stanford University Press, Stanford (1963) · Zbl 0109.06001 |
| [21] | Kuang, Y.: Delay differential equations with applications in population dynamics. Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston (1993) · Zbl 0777.34002 |
| [22] | Liu, B., Huang, L.: Boundedness of solutions for a class of retarded Liénard equation. J. Math. Anal. Appl. 286(2), 422-434 (2003) · Zbl 1044.34023 · doi:10.1016/S0022-247X(03)00455-4 |
| [23] | Liu, B., Huang, L.: Boundedness of solutions for a class of Liénard equations with a deviating argument. Appl. Math. Lett. 21(2), 109-112 (2008) · Zbl 1136.34328 · doi:10.1016/j.aml.2007.03.004 |
| [24] | Liu, C.J., Xu, S.L.: Boundedness of solutions of Liénard equations. J. Qingdao Univ. Nat. Sci. Ed. 11(3), 12-16 (1998) |
| [25] | Liu, Xin-Ge, Tang, Mei-Lan, Martin, Ralph R.: Periodic solutions for a kind of Liénard equation. J. Comput. Appl. Math. 219(1), 263-275 (2008) · Zbl 1152.34047 · doi:10.1016/j.cam.2007.07.024 |
| [26] | Liu, Z.R.: Conditions for the global stability of the Liénard equation. Acta Math. Sinica 38(5), 614-620 (1995) · Zbl 0838.34064 |
| [27] | Luk, W.S.: Some results concerning the boundedness of solutions of Lienard equations with delay. SIAM J. Appl. Math. 30(4), 768-774 (1976) · Zbl 0347.34055 · doi:10.1137/0130069 |
| [28] | Lyapunov, A.M.: Stability of motion. Academic Press, London (1966) · Zbl 0161.06303 |
| [29] | Malyseva, I.A.: Boundedness of solutions of a Liénard differential equation. Differetial’niye Uravneniya 15(8), 1420-1426 (1979) · Zbl 0497.34027 |
| [30] | Murean, M.: Boundedness of solutions for Liénard type equations. Mathematica 40(63), 243-257 (1998) · Zbl 1281.34043 |
| [31] | Valdés, Nápoles: Juan E., Boundedness and global asymptotic stability of the forced Liénard equation. (Spanish) Rev. Un. Mat. Argentina 41(4), 47-59 (2000) · Zbl 1028.34050 |
| [32] | Ponzo, P.J., Wax, N.: Stability, singular perturbations, and the vector Liénard equation. IEEE Trans. Autom. Control AC 17(4), 563-565 (1972) · Zbl 0262.93029 · doi:10.1109/TAC.1972.1100056 |
| [33] | Sugie, J.: On the boundedness of solutions of the generalized Liénard equation without the signum condition. Nonlinear Anal. 11(12), 1391-1397 (1987) · Zbl 0648.34036 · doi:10.1016/0362-546X(87)90091-5 |
| [34] | Sugie, J., Amano, Y.: Global asymptotic stability of non-autonomous systems of Liénard type. J. Math. Anal. Appl. 289(2), 673-690 (2004) · Zbl 1047.34062 · doi:10.1016/j.jmaa.2003.09.023 |
| [35] | Sugie, J., Chen, D.L., Matsunaga, H.: On global asymptotic stability of systems of Liénard type. J. Math. Anal. Appl. 219(1), 140-164 (1998) · Zbl 0913.34043 · doi:10.1006/jmaa.1997.5773 |
| [36] | Tunç, C.: Some new stability and boundedness results on the solutions of the nonlinear vector differential equations of second order. Iran. J. Sci. Technol. Trans. A Sci. 30(2), 213-221 (2006) |
| [37] | Tunç, C.: A new boundedness theorem for a class of second order differential equations. Arab. J. Sci. Eng. Sect. A Sci. 33(1), 83-92 (2008) · Zbl 1195.34051 |
| [38] | Tunç, C.: Some stability and boundedness results to nonlinear differential equations of Liénard type with finite delay. J. Comput. Anal. Appl. 11(4), 711-727 (2009) · Zbl 1176.34089 |
| [39] | Tunç, C.: New stability and boundedness results of solutions of Liénard type equations with multiple deviating arguments. J. Contemp. Math. Anal. 45(3), 47-56 (2010) · Zbl 1200.34086 |
| [40] | Tunç, C.: A note on boundedness of solutions to a class of non-autonomous differential equations of second order. Appl. Anal. Discrete Math. 4, 361-372 (2010) · Zbl 1299.34128 · doi:10.2298/AADM100601026T |
| [41] | Tunç, C.: Boundedness results for solutions of certain nonlinear differential equations of second order. J. Indones. Math. Soc. 16(2), 115-127 (2010) · Zbl 1232.34057 |
| [42] | Tunç, C.: Stability and boundedness of solutions of non-autonomous differential equations of second order. J. Comput. Anal. Appl. 13(6), 1067-1074 (2011) · Zbl 1227.34054 |
| [43] | Tunç, C.: Stability and boundedness in multi delay Lienard equation. Filomat. 27(3), 437-447 (2013) · Zbl 1324.34145 |
| [44] | Tunç, C., Tunç, E.: On the asymptotic behavior of solutions of certain second-order differential equations. J. Franklin Inst. 344(5), 391-398 (2007) · Zbl 1269.34057 · doi:10.1016/j.jfranklin.2006.02.011 |
| [45] | Utz, W.R.: Boundedness and periodicity of solutions of the generalized Liénard equation. Ann. Mat. Pura Appl. 42(4), 313-324 (1956) · Zbl 0071.30401 · doi:10.1007/BF02411883 |
| [46] | Yang, Q.G.: Boundedness and global asymptotic behavior of solutions to the Liénard equation. J. Syst. Sci. Math. Sci. 19(2), 211-216 (1999) · Zbl 0958.34030 |
| [47] | Yu, Y., Zhao, C.: Boundedness of solutions for a Liénard equation with multiple deviating arguments. Electron. J. Differ. Equ. 14, 5 (2009) · Zbl 1171.34339 |
| [48] | Yoshizawa, T.: Stability theory by Liapunov’s second method. Publications of the Mathematical Society of Japan, No. 9 The Mathematical Society of Japan, Tokyo (1966) · Zbl 0144.10802 |
| [49] | Yoshizawa, T.: Asymptotic behaviors of solutions of differential equations. Differential equations, qualitative theory, Vol. I, II (Szeged, 1984), 1141-1164, Colloq. Math. Soc. János Bolyai, 47, North-Holland, (1987) · Zbl 0585.34038 |
| [50] | Zhang, B.: On the retarded Liénard equation. Proc. Amer. Math. Soc. 115(3), 779-785 (1992) · Zbl 0756.34075 |
| [51] | Zhang, B.: Boundedness and stability of solutions of the retarded Liénard equation with negative damping. Nonlinear Anal. 20(3), 303-313 (1993) · Zbl 0773.34056 · doi:10.1016/0362-546X(93)90165-O |
| [52] | Zhang, B.: Necessary and sufficient conditions for boundedness and oscillation in the retarded Liénard equation. J. Math. Anal. Appl. 200(2), 453-473 (1996) · Zbl 0855.34090 · doi:10.1006/jmaa.1996.0216 |
| [53] | Zhang, X.S., Yan, W.P.: Boundedness and asymptotic stability for a delay Liénard equation. Math. Practice Theory 30(4), 453-458 (2000) · Zbl 1493.34196 |
| [54] | Zhao, L.: Boundedness and convergence for the non-Liénard type differential equation. Acta Math. Sci. Ser. B Engl. Ed. 27(2), 338-346 (2007) · Zbl 1125.34028 · doi:10.1016/S0252-9602(07)60034-4 |
| [55] | Zhou, J., Liu, Z.R.: The global asymptotic behavior of solutions for a nonautonomous generalized Liénard system. J. Math. Res. Expos. 21(3), 410-414 (2001) · Zbl 1002.34038 |
| [56] | Zhou, J., Xiang, L.: On the stability and boundedness of solutions for the retarded Liénard-type equation. Ann. Differ. Equ. 15(4), 460-465 (1999) · Zbl 0964.34064 |
| [57] | Zhou, X.-F., Jiang, W.: Stability and boundedness of retarded Liénard-type equation. Chin. Quart. J. Math. 18(1), 7-12 (2003) · Zbl 1058.34098 |
| [58] | Wei, J., Huang, Q.: Global existence of periodic solutions of Liénard equations with finite delay. Dynam. Contin. Discrete Impuls. Syst. 6(4), 603-614 (1999) · Zbl 0953.34059 |
| [59] | Wiandt, T.: On the boundedness of solutions of the vector Liénard equation. Dynam. Syst. Appl. 7(1), 141-143 (1998) · Zbl 0901.34041 |
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