Sufficient conditions for the boundedness and square integrability of Solutions of fourth-order differential equations
DOI:
https://doi.org/10.4067/S0716-09172016000100004Keywords:
Boundedness, stability, Lyapunov functional, fourth-order differential equations, square integrable, acotamiento, estabilidad, funcional de Lyapunov, ecuaciones diferenciales de cuarto orden, función de cuadrado integrable.Abstract
Sufficient conditions for the boundedness and square integrability of solutions and their derivatives of certain fourth order nonlin-ear differential equation are given by means of the Lyapunov’s second method. Our results obtained in this work, generalize existing results on fourth order nonlinear differential equations in the literature. For illustration, an example is also given.References
[1] Afuwape, A .U. and Adesina, O. A.; Frequency-domain approach to stability and periodic solutions of certain fourth-order non-linear differential equations. Nonlinear Stud. 12, No. 3, pp. 259-269, (2005).
[2] Andres, J., and Vlcek, V.; On the existence of square integrable solutions and their derivatives to fourth and fifth order differential equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 28, No. 1, pp. 65—86, (1989).
[3] Burton T. A., Stability and periodic solutions of ordinary and functional differential equations .Mathematics in science and engineering, Volume 178, Academic Press, Inc, (1985).
[4] Burton T.A., Volterra Integraln and Differential Equations, Mathematics in Science and Engineering Vol. (202), 2nd edition, (2005).
[5] Chin, P. S. M.; Stability results for the solutions of certain fourth-order autonomous differential equations. Internat. J. Control. 49, No. 4, pp. 1163-1173, (1989).
[6] Ezeilo, J. O. C.; On the Boundedness and the Stability of Solution of some Fourth Order Equations, J. Math. Anal. Appl. 5, pp. 136-146, (1962).
[7] Ezeilo, J. O. C.; A Stability result for Solutions of a Certain Fourth Order Diffrential Equations. J.London Math. Soc. 37, pp. 28-32, (1962).
[8] Harrow, M.; A Stability result for Solutions of a Certain Fourth Order Homogeneous Differential Equations, J. London Math. Soc. 42, pp. 51-56, (1967).
[9] Harrow, M.; On the Boundedness and the Stability of Solutions of some Differential Equations of the Fourth Order, SIAM, J. Math. Anal. 1, pp. 27-32, (1970).
[10] Hara, T. On the asymptotic behavior of the solutions of some third and fourth order non-autonomous differential equations. Publ. RIMS, Kyoto Univ. 9, pp. 649-673, (1974).
[11] Hara, T.; On the asymptotic behavior of solutions of some third order ordinary differential equations. Proc. Japan Acad., 47 (1971).
[12] Omeike, M. O.; Boundedness of solutions of the fourth order differential equation with oscillatory restoring and forcing terms. An. Stiint¸ Univ. Al. I. Cuza Iasi. Mat. (N.S.) 54, No. 1, pp. 187-195, (2008).
[13] Remili, M. and Oudjedi, D. L.; Uniform stability and boundedness of a kind of third order delay differential equations. Bull. Comput. Appl. Math., Vol 2, No. 1, (2014).
[14] Remili, M. and Oudjedi, D. L.; 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).
[15] Remili, M. and Beldjerd, D.; On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo, Vol 63, No 3, (2014).
[16] Shair, A.; Asymptotic properties of linear fourth order differential equations. American mathematical society volume 59. No 1, august (1976).
[17] Tiryaki,A. and Tunc,C.; Boundedness and the Stability Properties of Solutions of Certain Fourth Order Differential Equations via the Intrinsic Method, Analysis, 16, pp. 325-334, (1996).
[18] Tunc, C.; A Note on the Stability and Boundedness Results of Certain Fourth Order Differential Equations, Applied Mathematics and Computation, 155, No. 3, pp. 837-843, (2004).
[19] Tunc, C.; Some Stability and Boundedness Results for the Solutions of Certain Fourth Order Differential Equations, Acta Univ. Palacki Olomouc. Fac. Rerum Natur. Math. 44, pp. 161-171, (2005).
[20] Tunc, C.; An ultimate Boundedness Result for a Certain System of Fourth Order Nonlinear Differential Equations, Differential Equations and Applications, Vol. 5, pp. 16-174, (2005).
[21] Tunc, C. and Tiryaki,A.; On the Boundedness and the Stability Results for the Solutions of Certain Fourth Order Differential Equations via the Intrinsic Method, Applied Mathematics and Mechanics, 17, No. 11, pp. 1039-1049, (1996).
[22] Tiryaki. A.; Tunc, C. Constructing Lyapunov functions for certain fourthorder autonomous differential equations. Indian J. Pure Appl. Math. 26, No. 3, pp. 225-232, (1995).
[23] Tunc, C.; Stability and boundedness of solutions to certain fourth order differential equations. Electronic Journal of Differential Equations, Vol. 2006, No. 35, pp. 1-10, (2006).
[24] Tunc, C.; Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order. and Computational and Applied Mathematics, Volume 26, N. 1, pp. 1-17, (2007).
[25] Tunc, C., and Ates, M.; Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms. Discrete Dynamics in Nature and Society Volume, pp. 1-6, (2013).
[26] Vlcek, V.; On the boundedness of solutions of a certain fourth-order nonlinear differential equation. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 27, No. 1, pp. 273—288, (1988).
[27] Yoshizawa, T.; Stability Theory by Liapunovs Second Method, The Mathematical Society of Japan, Tokyo, (1966).
[2] Andres, J., and Vlcek, V.; On the existence of square integrable solutions and their derivatives to fourth and fifth order differential equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 28, No. 1, pp. 65—86, (1989).
[3] Burton T. A., Stability and periodic solutions of ordinary and functional differential equations .Mathematics in science and engineering, Volume 178, Academic Press, Inc, (1985).
[4] Burton T.A., Volterra Integraln and Differential Equations, Mathematics in Science and Engineering Vol. (202), 2nd edition, (2005).
[5] Chin, P. S. M.; Stability results for the solutions of certain fourth-order autonomous differential equations. Internat. J. Control. 49, No. 4, pp. 1163-1173, (1989).
[6] Ezeilo, J. O. C.; On the Boundedness and the Stability of Solution of some Fourth Order Equations, J. Math. Anal. Appl. 5, pp. 136-146, (1962).
[7] Ezeilo, J. O. C.; A Stability result for Solutions of a Certain Fourth Order Diffrential Equations. J.London Math. Soc. 37, pp. 28-32, (1962).
[8] Harrow, M.; A Stability result for Solutions of a Certain Fourth Order Homogeneous Differential Equations, J. London Math. Soc. 42, pp. 51-56, (1967).
[9] Harrow, M.; On the Boundedness and the Stability of Solutions of some Differential Equations of the Fourth Order, SIAM, J. Math. Anal. 1, pp. 27-32, (1970).
[10] Hara, T. On the asymptotic behavior of the solutions of some third and fourth order non-autonomous differential equations. Publ. RIMS, Kyoto Univ. 9, pp. 649-673, (1974).
[11] Hara, T.; On the asymptotic behavior of solutions of some third order ordinary differential equations. Proc. Japan Acad., 47 (1971).
[12] Omeike, M. O.; Boundedness of solutions of the fourth order differential equation with oscillatory restoring and forcing terms. An. Stiint¸ Univ. Al. I. Cuza Iasi. Mat. (N.S.) 54, No. 1, pp. 187-195, (2008).
[13] Remili, M. and Oudjedi, D. L.; Uniform stability and boundedness of a kind of third order delay differential equations. Bull. Comput. Appl. Math., Vol 2, No. 1, (2014).
[14] Remili, M. and Oudjedi, D. L.; 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).
[15] Remili, M. and Beldjerd, D.; On the asymptotic behavior of the solutions of third order delay differential equations. Rend. Circ. Mat. Palermo, Vol 63, No 3, (2014).
[16] Shair, A.; Asymptotic properties of linear fourth order differential equations. American mathematical society volume 59. No 1, august (1976).
[17] Tiryaki,A. and Tunc,C.; Boundedness and the Stability Properties of Solutions of Certain Fourth Order Differential Equations via the Intrinsic Method, Analysis, 16, pp. 325-334, (1996).
[18] Tunc, C.; A Note on the Stability and Boundedness Results of Certain Fourth Order Differential Equations, Applied Mathematics and Computation, 155, No. 3, pp. 837-843, (2004).
[19] Tunc, C.; Some Stability and Boundedness Results for the Solutions of Certain Fourth Order Differential Equations, Acta Univ. Palacki Olomouc. Fac. Rerum Natur. Math. 44, pp. 161-171, (2005).
[20] Tunc, C.; An ultimate Boundedness Result for a Certain System of Fourth Order Nonlinear Differential Equations, Differential Equations and Applications, Vol. 5, pp. 16-174, (2005).
[21] Tunc, C. and Tiryaki,A.; On the Boundedness and the Stability Results for the Solutions of Certain Fourth Order Differential Equations via the Intrinsic Method, Applied Mathematics and Mechanics, 17, No. 11, pp. 1039-1049, (1996).
[22] Tiryaki. A.; Tunc, C. Constructing Lyapunov functions for certain fourthorder autonomous differential equations. Indian J. Pure Appl. Math. 26, No. 3, pp. 225-232, (1995).
[23] Tunc, C.; Stability and boundedness of solutions to certain fourth order differential equations. Electronic Journal of Differential Equations, Vol. 2006, No. 35, pp. 1-10, (2006).
[24] Tunc, C.; Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order. and Computational and Applied Mathematics, Volume 26, N. 1, pp. 1-17, (2007).
[25] Tunc, C., and Ates, M.; Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms. Discrete Dynamics in Nature and Society Volume, pp. 1-6, (2013).
[26] Vlcek, V.; On the boundedness of solutions of a certain fourth-order nonlinear differential equation. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Vol. 27, No. 1, pp. 273—288, (1988).
[27] Yoshizawa, T.; Stability Theory by Liapunovs Second Method, The Mathematical Society of Japan, Tokyo, (1966).
Published
2017-03-23
How to Cite
[1]
M. Remili and M. Rahmane, “Sufficient conditions for the boundedness and square integrability of Solutions of fourth-order differential equations”, Proyecciones (Antofagasta, On line), vol. 35, no. 1, pp. 41-61, 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.
