Dördüncü mertebeden sınırlı gecikmeli nonlineer diferansiyel denklemlerin çözümlerinin kararlılığı ve sınırlılığı
Bu makalede Lyapunov’un ikinci metodu kullanılarak dördüncü mertebeden otonom olmayan değişken gecikmeli diferansiyel denklemlerin çözümlerinin düzgün asimptotik kararlılığı ve sınırlılığı için yeterli şartları veririz. Elde edilen sonuçlar literatürdeki sonuçları tamamlar, kapsar ve geliştirir.
Stability and boundedness of solutions of nonlinear fourth order differential equations with bounded delay
In this paper, we determine sufficient conditions for the boundedness, uniformly asymtotically stability ofthe solutions to a certain fourth-order non-autonomous differential equations with bounded delay byconsidering second method of Lyapunov. The results obtain essentially improve, include and complementthe consequences in the current literature.
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- A. M. A. Abou-Ela-Ela and A. I. Sadek, "A
stability result for certain fourth order
differential equations," Ann. Differential
Equations, vol. 1, no. 6, pp. 1-9, 1990.
- O. A. Adesina and B. S. Ogundare, "Some
new stability and boundedness results on a
certainfourth order nonlinear differential
equation.," Nonlinear Stud, vol. 3, no. 19,
pp. 359-369, 2012.
- H. Bereketoğlu, "Asymptotic stability in a
fourth order delay differential equation.,"
Dynam. Systems Appl, vol. 1, no. 7, pp. 105-
115, 1998.
- T. A. Burton, Stability and periodic solutions
of ordinary and functional differential
equations. Mathematics in science and
engineering, 1985.
- L. Elsgolts, Introduction to the Theory of
Differential Equations with Deviating
Arguments. Translated from the Russian by
Robert J. McLaughlin Holden-Day, Inc. San
Francisco Calif.-London-Amsterdam, 1966.
- J. O. C. Ezeilo and H. O. Tejumola, "On the
boundedness and the stability properties of
solutions of certain fourth order differential
equations," Ann. Mat. Pura Appl., vol. 4, no.
95, pp. 131-145, 1973.
- J. O. C. Ezeilo, "Stability results for the
solutions of some third and fourth order
differential equations," Ann. Mat. Pura
Appl., vol. 4, no. 66, pp. 233-249, 1964.
- T. Hara, "On the asymptotic behavior of the
solutions of some third and fourth order
nonautonomous differential equations,"
Publ. RIMS, Kyoto Univ., no. 9, pp. 649-673,
1974.
- M. Harrow, "Further results on the
boundedness and the stability of solutions of
some differential equations of the fourth order," SIAM J. Math. Anal., no. 1, pp. 189-
194, 1970.
- M. Harrow, "A Stability result for Solutions
of a Certain Fourth Order Homogeneous
Differential Equations," J. London Math.
Soc., no. 42, pp. 51-56, 1967.
- Z. Körpınar. "Kesirli Klein-Gordon
denklemi için residual power seri metodu. "
SAÜ Fen Bilimleri Enstitüsü Dergisi, 21 (3),
285-293, 2017.
- N. N. Krasovskii, "On the stability in the
large of the solution of a nonlinear system of
differential equations (Russian)," Prikl. Mat.
Meh., no. 18, pp. 735-737, 1954.
- A. M. Lyapunov, The general problem of the
stability of motion. Translated from Edouard
Davaux's French translation (1907) of the
1892 Russian original and edited by A. T.
Fuller., London: Taylor & Francis, Ltd.,
1992.
- MM Al Qurashi, ZS Korpinar, M. Inc,
"Approximate solutions of bright and dark
optical solitons in birefrigent fibers, " OptikInternational Journal for Light and Electron
Optics, 140, 45-61, 2017.
- M. Rahmane and M. Remili, "On Stability
and Boundedness of Solutions of Certain
Non Autonomous Fourth-Order Delay
Differential Equations," Acta Universitatis
Matthiae Belii, series Mathematics Issue,
no. 17-30, 2015.
- R. Reissig, G. Sansone and R. Conti, Nonlinear Differential Equations of Higher
Order. Translated from the German.,
Leyden: Noordhoff International Publishing,
1974.
- M. Remili and M. Rahmane, "Boundedness
and square integrability of solutions of
nonlinear fourth-order differential
equations," Nonlinear Dynamics and
Systems Theory, vol. 2, no. 16, pp. 192-205,
2016.
- M. Remili and M. Rahmane, "Sufficient
conditions for the boundedness and square
integrability of solutions of fourth-order
differential equations," Proyecciones
Journal of Mathematics., vol. 1, no. 35, pp.
41-61, 2016.
- A. Shair, "Asymptotic properties of linear
fourth order differential equations,"
American mathematical society, vol. 1, no.
59, pp. 45-51, 1976.
- A. S. C. Sinha, "On stability of solutions of
some third and fourth order delaydifferential equations," Information and
Control, no. 23, pp. 165-172, 1973.
- H. O. Tejumola, "Further results on the
boundedness and the stability of certain
fourth order differential equations," Atti
Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat.
Natur., vol. 8, no. 52, pp. 16-23, 1972.
- C. Tunç, "Some remarks on the stability and
boundedness of solutions of certain
differential equations of fourth-order,"
Computational and Applied Mathematics,
vol. 1, no. 26, pp. 1-17, 2007.
- C. Tunç, "Some stability and boundedness
results for the solutions of certain fourth
order differential equations," Acta Univ.
Palack. Olomuc. Fac. Rerum Natur. Math.,
no. 44, pp. 161-171, 2005.
- C. Tunç, "Some stability results for the
solutions of certain fourth order delay
differential equations," Differential
Equations and Applications, no. 4, pp. 165-
174, 2004.
- C. Tunç, "Stability and boundedness of
solutions to certain fourth order differential
equations," Electronic Journal of
Differential Equations, no. 35, pp. 1-10,
2006.
- C. Tunç, "Stability and boundedness of
solutions to certain fourth order differential
equations," Electron. J. Differential
Equations, no. 35, p. 10pp., 2006.
- X. Wu and K. Xiong, "Remarks on stability
results for the solutions of certain fourthorder autonomous differential equations,"
Internat. J. Control, vol. 2, no. 69, p. 353,
1998.
- T. Yoshizawa, Stability Theory by
Liapunovs Second Method, Tokyo,: The
Mathematical Society of Japan, 1966.