On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order

Affiliation(s)

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.

Department of Mathematics, Faculty of Science, New Valley Branch, Assiut University, New Valley, El-Khargah, Egypt.

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.

Department of Mathematics, Faculty of Science, New Valley Branch, Assiut University, New Valley, El-Khargah, Egypt.

ABSTRACT

The main purpose of this paper is to investigate global asymptotic
stability of the zero solution of the fifth-order nonlinear delay differential
equation on the following form

By constructing a Lyapunov functional, sufficient conditions for the stability of the zero solution of this equation are established.

Cite this paper

Abou-El-Ela, A. , Sadek, A. , Mahmoud, A. and Taie, R. (2014) On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order.*Advances in Pure Mathematics*, **4**, 357-367. doi: 10.4236/apm.2014.48046.

Abou-El-Ela, A. , Sadek, A. , Mahmoud, A. and Taie, R. (2014) On the Stability of Solutions of Nonlinear Functional Differential Equation of the Fifth-Order.

References

[1] Adesina, O.A. (2007) Convergence Results for Solutions of a Certain Fifth-Order Nonlinear Differential Equation. Nonlinear Studies, 14, 289-300.

[2] Bereketoglu, H. (1998) Asymptotic Stability of a Fourth-Order Delay Differential Equation. Dynamic Systems an Applications, 7, 105-115.

[3] Burton, T.A. (1983) Volterra Integral and Differential Equations. Academic Press, New York.

[4] Burton, T.A. (1985) Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, New York.

[5] Chukwu, E.N. (1978) On the Boundedness and the Existence of a Periodic Solution of Some Nonlinear Third-Order Delay Differential Equation. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 64, 440-447.

[6] Harrow, M. (1967) A Stability Result for Solutions of Certain Fourth-Order Homogeneous Differential Equations. Journal London Mathematical Society, 42, 51-56. http://dx.doi.org/10.1112/jlms/s1-42.1.51

[7] Harrow, M. (1970) On the Boundedness and the Stability of Solutions of Some Differential Equations of the Fourth-Order. SIAM Journal on Mathematical Analysis, 1, 27-32. http://dx.doi.org/10.1137/0501002

[8] Hale, J.K. (1977) Theory of Functional Differential Equations. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-1-4612-9892-2

[9] Kolmanovskii, V. and Myshkis, A. (1999) Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-017-1965-0

[10] Makay, G. (1994) On the Asymptotic Stability of the Solutions of Functional Differential Equations with Infinite Delay. Journal of Differential Equations, 108, 139-151. http://dx.doi.org/10.1006/jdeq.1994.1029

[11] Okoronkwo, E.O. (1989) On Stability and Boundedness of Solutions of a Certain Fourth-Order Delay Differential Equation. International Journal of Mathematics and Mathematical Sciences, 12, 589-602.

http://dx.doi.org/10.1155/S0161171289000724

[12] Reissing, R., Sansone, E. and Conti, R. (1974) Non-Linear Differential Equations of Higher-Order. Noordhoof, Toom.

[13] Sadek, A.I. (2003) Stability and Boundedness of a Kind of Third-Order Delay Differential System. Applied Mathematics Letters, 16, 657-662. http://dx.doi.org/10.1016/S0893-9659(03)00063-6

[14] Sadek, A.I. (2004) On the Stability of Solutions of Certain Fourth-Order Delay Differential Equations. Applied Mathe-matics and Computation, 148, 587-597. http://dx.doi.org/10.1016/S0096-3003(02)00925-6

[15] Sadek, A.I. (2005) On the Stability of Solutions of Some Non-Autonomous Delay Differential Equations of the Third-Order. Asymptotic Analysis, 43, 1-7.

[16] Li, S.L. and Wen, L.Z. (1987) Functional Differential Equations. Hunan Science and Technology Press, Changsha.

[17] Sinha, A.S.C. (1973) On Stability of Solutions of Some Third and Fourth-Order Delay Differential Equations. Information and Control, 23, 165-172. http://dx.doi.org/10.1016/S0019-9958(73)90651-7

[18] Tejumola, H.O. and Tchegnani, B. (2000) Stability, Boundedness and Existence of Periodic Solutions of Some Third and Fourth-Order Nonlinear Delay Differential Equations. Journal of the Nigerian Mathematics Society, 19, 9-19.

[19] Tunc, C. (2002) A Study of the Stability and Boundedness of the Solutions of Nonlinear Differential Equations of the Fifth-Order. Indian Journal of Pure and Applied Mathematics, 33, 519-529.

[20] Tunc, C. (2005) Some Stability Results for the Solutions of Certain Fourth-Order Delay Differential Equations. Differential Equations & Applications, 4, 165-174.

[21] Tunc, C. (2006) On the Stability of Solutions of Certain Fourth-Order Delay Differential Equations. Applied Mathematics and Mechanics, 27, 1141-1148.

[22] Tunc, C. (2008) On the Stability of Solutions to a Certain Fourth-Order Delay Differential Equation. Nonlinear Dynamics, 51, 71-81. http://dx.doi.org/10.1007/s11071-006-9192-z

[23] Tunc, C. (2010) On Asymptotic Stability of Solutions of Fifth-Order Nonlinear Differential Equations with Delay. Functional Differential Equations, 17, 355-370.

[24] Tunc, C. (2013) New Results on the Stability and Boundedness of Nonlinear Differential Equations of Fifth-Order with Multiple Deviating Arguments. Bulletin of the Malaysian Mathematical Sciences Society, 36, 671-682.

[25] Yoshizawa, T. (1966) Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan.

[26] Qin, Y., Liu, Y. and Wang, L. (1966) Stability of Motion for Dynamic Systems with Delay. Academic Press, Beijing.

[27] Zhu, Y. (1992) On Stability, Boundedness and Existence of Periodic Solution of a Kind of Third-Order Nonlinear Delay Differential System. Annals of Differential Equations, 8, 249-259.

[28] Lyapunov, A.M. (1966) Stability of Motion. Academic Press, London.

[29] Tunc, C. (2010) On the Stability of Solutions of Nonlinear Differential Equations of Fifth-Order with Delay. Mathematical Communications, 15, 261-272.

[30] Abou-El-Ela, A.M.A., Sadek, A.I. and Mahmoud, A.M. (2011) On the Stability of Solutions of a Certain Nonlinear Delay Differential Equation of the Fifth-Order. Proceedings of the 3rd Scientific Conference for Young Researchers, Assiut, 19-20 April 2011.

[31] Burton, T.A. and Hering, G. (1994) Lyapunov Theory for Functional Differential Equations. Rocky Mountain Journal of Mathematics, 24, 3-17. http://dx.doi.org/10.1216/rmjm/1181072449

[32] Hale, J.K. (1965) Sufficient Conditions for Stability and Instability of Autonomous Functional-Differential Equations. Journal of Differential Equations, 1, 452-482. http://dx.doi.org/10.1016/0022-0396(65)90005-7

[1] Adesina, O.A. (2007) Convergence Results for Solutions of a Certain Fifth-Order Nonlinear Differential Equation. Nonlinear Studies, 14, 289-300.

[2] Bereketoglu, H. (1998) Asymptotic Stability of a Fourth-Order Delay Differential Equation. Dynamic Systems an Applications, 7, 105-115.

[3] Burton, T.A. (1983) Volterra Integral and Differential Equations. Academic Press, New York.

[4] Burton, T.A. (1985) Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, New York.

[5] Chukwu, E.N. (1978) On the Boundedness and the Existence of a Periodic Solution of Some Nonlinear Third-Order Delay Differential Equation. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 64, 440-447.

[6] Harrow, M. (1967) A Stability Result for Solutions of Certain Fourth-Order Homogeneous Differential Equations. Journal London Mathematical Society, 42, 51-56. http://dx.doi.org/10.1112/jlms/s1-42.1.51

[7] Harrow, M. (1970) On the Boundedness and the Stability of Solutions of Some Differential Equations of the Fourth-Order. SIAM Journal on Mathematical Analysis, 1, 27-32. http://dx.doi.org/10.1137/0501002

[8] Hale, J.K. (1977) Theory of Functional Differential Equations. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-1-4612-9892-2

[9] Kolmanovskii, V. and Myshkis, A. (1999) Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-017-1965-0

[10] Makay, G. (1994) On the Asymptotic Stability of the Solutions of Functional Differential Equations with Infinite Delay. Journal of Differential Equations, 108, 139-151. http://dx.doi.org/10.1006/jdeq.1994.1029

[11] Okoronkwo, E.O. (1989) On Stability and Boundedness of Solutions of a Certain Fourth-Order Delay Differential Equation. International Journal of Mathematics and Mathematical Sciences, 12, 589-602.

http://dx.doi.org/10.1155/S0161171289000724

[12] Reissing, R., Sansone, E. and Conti, R. (1974) Non-Linear Differential Equations of Higher-Order. Noordhoof, Toom.

[13] Sadek, A.I. (2003) Stability and Boundedness of a Kind of Third-Order Delay Differential System. Applied Mathematics Letters, 16, 657-662. http://dx.doi.org/10.1016/S0893-9659(03)00063-6

[14] Sadek, A.I. (2004) On the Stability of Solutions of Certain Fourth-Order Delay Differential Equations. Applied Mathe-matics and Computation, 148, 587-597. http://dx.doi.org/10.1016/S0096-3003(02)00925-6

[15] Sadek, A.I. (2005) On the Stability of Solutions of Some Non-Autonomous Delay Differential Equations of the Third-Order. Asymptotic Analysis, 43, 1-7.

[16] Li, S.L. and Wen, L.Z. (1987) Functional Differential Equations. Hunan Science and Technology Press, Changsha.

[17] Sinha, A.S.C. (1973) On Stability of Solutions of Some Third and Fourth-Order Delay Differential Equations. Information and Control, 23, 165-172. http://dx.doi.org/10.1016/S0019-9958(73)90651-7

[18] Tejumola, H.O. and Tchegnani, B. (2000) Stability, Boundedness and Existence of Periodic Solutions of Some Third and Fourth-Order Nonlinear Delay Differential Equations. Journal of the Nigerian Mathematics Society, 19, 9-19.

[19] Tunc, C. (2002) A Study of the Stability and Boundedness of the Solutions of Nonlinear Differential Equations of the Fifth-Order. Indian Journal of Pure and Applied Mathematics, 33, 519-529.

[20] Tunc, C. (2005) Some Stability Results for the Solutions of Certain Fourth-Order Delay Differential Equations. Differential Equations & Applications, 4, 165-174.

[21] Tunc, C. (2006) On the Stability of Solutions of Certain Fourth-Order Delay Differential Equations. Applied Mathematics and Mechanics, 27, 1141-1148.

[22] Tunc, C. (2008) On the Stability of Solutions to a Certain Fourth-Order Delay Differential Equation. Nonlinear Dynamics, 51, 71-81. http://dx.doi.org/10.1007/s11071-006-9192-z

[23] Tunc, C. (2010) On Asymptotic Stability of Solutions of Fifth-Order Nonlinear Differential Equations with Delay. Functional Differential Equations, 17, 355-370.

[24] Tunc, C. (2013) New Results on the Stability and Boundedness of Nonlinear Differential Equations of Fifth-Order with Multiple Deviating Arguments. Bulletin of the Malaysian Mathematical Sciences Society, 36, 671-682.

[25] Yoshizawa, T. (1966) Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan.

[26] Qin, Y., Liu, Y. and Wang, L. (1966) Stability of Motion for Dynamic Systems with Delay. Academic Press, Beijing.

[27] Zhu, Y. (1992) On Stability, Boundedness and Existence of Periodic Solution of a Kind of Third-Order Nonlinear Delay Differential System. Annals of Differential Equations, 8, 249-259.

[28] Lyapunov, A.M. (1966) Stability of Motion. Academic Press, London.

[29] Tunc, C. (2010) On the Stability of Solutions of Nonlinear Differential Equations of Fifth-Order with Delay. Mathematical Communications, 15, 261-272.

[30] Abou-El-Ela, A.M.A., Sadek, A.I. and Mahmoud, A.M. (2011) On the Stability of Solutions of a Certain Nonlinear Delay Differential Equation of the Fifth-Order. Proceedings of the 3rd Scientific Conference for Young Researchers, Assiut, 19-20 April 2011.

[31] Burton, T.A. and Hering, G. (1994) Lyapunov Theory for Functional Differential Equations. Rocky Mountain Journal of Mathematics, 24, 3-17. http://dx.doi.org/10.1216/rmjm/1181072449

[32] Hale, J.K. (1965) Sufficient Conditions for Stability and Instability of Autonomous Functional-Differential Equations. Journal of Differential Equations, 1, 452-482. http://dx.doi.org/10.1016/0022-0396(65)90005-7