ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
и
Процессы Управления

Uniform Stability, Boundedness and Asymptotic Behaviour of Solutions of Some Third Order Nonlinear Delay Differential Equations

Автор(ы):

A. T. Ademola

Department of Mathematics
University of Ibadan,
Ibadan, Nigeria

ademola672000@yahoo.com

P. O. Arawomo

Department of Mathematics
University of Ibadan,
Ibadan, Nigeria

womopeter@gmail.com

O. M. Ogunlaran

Department of Mathematics and Statistics
Bowen University, Iwo,
Osun State, Nigeria

dothew2002@yahoo.com

E. A. Oyekan

Department of Mathematics and Statistics
Bowen University, Iwo,
Osun State, Nigeria

shalomfa@yahoo.com

Аннотация:

In this paper sufficient conditions for uniform asymptotic stability of the trivial solutions, uniform boundedness, uniform ultimate boundedness and asymptotic behaviour of solutions of some third order nonlinear neutral delay differential equations are established. We employ Lyapunov's direct method by constructing a complete Lyapunov functional to obtain the results. Recent results on third order nonlinear delay differential equations are particular cases of our results.

Ссылки:

  1. Ademola, A. T. and Arawomo, P. O.; Boundedness and asymptotic behaviour of solutions of a nonlinear differential equation of the third order. Afr. Mat. 23 (2012), no. 2, 261 - 271
  2. Ademola, A. T. and Arawomo, P. O.; Stability and ultimate boundedness of solutions to certain third-order differential equations. Applied Mathematics E-Notes. 10(2010), 61-69
  3. Ademola, A. T. and Arawomo, P. O.; Stability and uniform ultimate boundedness of solutions of a third-order differential equation. International Journal of Applied Mathematics. 23 no. 1, (2010) 11-22
  4. Ademola, A. T. and Arawomo, P. O.; Stability and uniform ultimate boundedness of solutions of some third order differential equations. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 27 (2011), 51-59
  5. Ademola, A. T. and Arawomo, P. O.; Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order. Mathematical Journal Okayama University 55 (2013), 157–166
  6. Ademola, A. T., Ogundiran, M. O., Arawomo, P. O. and Adesina, O. A.; Boundedness results for a certain third-order nonlinear differential equations. Appl. Math. Comput. 216 (2010), 3044-3049
  7. Ademola, A. T., Ogundiran, M. O., Arawomo, P. O. and Adesina, O. A.; Stability results for the solutions of a certain third-order nonlinear differential equation. Mathematical Sciences Research Journal MSRJ. Vol. 12, no. 6, (2008) 124-134
  8. Afuwape, A. U. and Omeike, M. O.; On the stability and boundedness of solutions of a kind of third order delay differential equations. Applied Mathematics and Computation, 200 (2008), 444-451
  9. Afuwape, A. U. and Omeike, M. O.; Stability and boundedness of solutions of a kind of third-order delay differential equations. Computational & Applied Mathematics, 29, N. 3, (2010) 329-342
  10. Ademola, A. T. and Arawomo, P. O.; Asymptotic behaviour of solutions of third order nonlinear differential equations. Acta Univ. Sapientiae, Mathematica, 3, 2 (2011) 197 - 211
  11. Bereketoglu, H. and Karakoc, F., Some results on boundedness and stability of a third order differential equation with delay, IASI, Tomul LI, s. I, Matematica, f. 2, 245-258, 2005
  12. Burton, T. A. and Hering, R.; Liapunov theory for functional differential equations. Rocky Mountain J. Math. 24 (1994), 3-17
  13. Burton, T. A. and Makey, G.; Asymptotic stability for functional differential equations. Acta Math. Hungar, 65 (1994), 243-251
  14. Burton, T. A.; Stability and periodic solutions of ordinary and functional differential equations. Mathematics in Science and Engineering, 178 Academic Press. Inc., Orlando, FL, 1985
  15. Burton, T. A.; Volterra integral and differential equations. New York: Academic Press, 1983
  16. Chukwu, E. N.; On boundedness of solutions of third order differential equations, Ann. Mat. Pura. Appl., 104 (4) (1975), 123-149
  17. Driver, R. D.; Ordinary and delay differential equations. Applied Mathematical Sciences, Springer-Verlag New York Inc. (1976)
  18. Ezeilo, J. O. C.; Further results for the solutions of a third-order differential equation, Proc. Camb. Phil. Soc., 59 (1963). 111-116
  19. Ezeilo, J. O. C.; A boundedness theorem for a certain third-order differential equation, Proc. London Math. Soc. (3) 13 (1963), 99-124
  20. Ezeilo, J. O. C.; On the stability of solutions of some third-order differential equations, J. London Math. Soc., 43 (1968), 161-167
  21. Hale, J. K.; Functional differential equations. Applied Mathematical Sciences, 3 Springer-verlag New York. Heidelberg. Berlin. (1971)
  22. Hale, J. K.; Theory of functional differential equations. Springer-verlag New York, (1977)
  23. Hara, T.; On the uniform ultimate boundedness of solutions of certain third-order differential equations, J. Math. Anal. Appl. (2) 80 (1981), 533-544
  24. Ogundare, B. S.; On the boundedness and stability results for the solutions of certain third-order nonlinear differential equations. Kragujevac J. Math. 29 (2006) 37-48
  25. Omeike, M. O.; New result in the ultimate boundedness of solutions of a third-order nonlinear ordinary differential equation, J. Inequal. Pure and Appl. Math., 9 (1) (2008), Art. 15, 8 pp
  26. Omeike, M. O.; New results on the asymptotic behavior of a third-order nonlinear differential equations, Differential Equations & Applications Vol. 2008 (2008) 1-13
  27. Omeike, M. O.; New results on the stability of solution of some non-autonomous delay differential equations of the third-order. Differential Equations and Control Processes 2010 1, (2010), 18-29
  28. Reissig, R., Sansone, G. and Conti, R.; Nonlinear differential equations of higher order, Noordhoff International Publishing Leyeden (1974)
  29. Rouche, N., Habets, N. and Laloy, M.; Stability theory by Liapunov's direct method. Applied Mathematical Sciences 22, Spriger-Verlag New York. Heidelberg. Berlin (1977)
  30. Sadek, A. I.; Stability and boundedness of a kind of third-order delay differential system. Applied Mathematics Letters 16 (2003) 657-662
  31. Swick, K. E.; Asymptotic behavior of the solutions of certain third-order differential equations, SIAM J. Appl., 19 N0. 1 (1970), 96-102
  32. Swick, K. E.; Boundedness and stability for a nonlinear third-order differential equation, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 56 (20), (1974), 859-865
  33. Tejumola, H. O.; A note on the boundedness of solutions of some nonlinear differential equations of the third-order, Ghana J. of Science, 11 (2), (1970), 117-118
  34. Tejumola, H. O.; A note on the boundedness and the stability of solutions of certain third-order differential equations. Ann. Math. Pura. Appl., 92, (1972) 65-75
  35. Tunc, C.; Boundedness of solutions of a third-order nonlinear differential equation, J. Inequal. Pure and Appl. Math., 6 (1) (2005), 1-6
  36. Tunc, C.; New results about stability and boundedness of solutions of certain nonlinear third-order delay differential equations. The Arabian Journal for Science and Engineering, 31. 2A (2006), 185-196
  37. Tunc, C.; On asymptotic stability of solutions to third-order nonlinear differential equations with retarded argument. Communications in Applied Analysis 11 (2007), 515-528
  38. Tunc, C.; On the asymptotic behavior of solutions of certain third-order nonlinear differential equations, J. Applied Math. and Stochastic Anal. 1 (2005) 29-35
  39. Tunc, C.; On the stability of solutions for non-autonomous delay differential equations of third-order, Iranian Journal of Science & Technology, Transaction A, Vol. 32, No. A4 (2008), 261-272
  40. Tunc, C.; Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differential Equations and Control Processes 2007 3 (2007), 1-12
  41. Tunc, C.; Some new results on the boundedness of solutions of a certain nonlinear differential equation of third-order. International J. of Nonlinear Science, vol. 7 No. 2 (2009) 246-256
  42. Tunc, C.; Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments. Electronic J. of Qualitative Theory of Differential Equations, 2010 1 (2010), 1-12
  43. Yamamoto, M.; Further results for the solutions of certain third-order non-autonomous differential equations. Proc. Japan Acad., No. 5, 45 (1973), 317-321
  44. Yao, H. and Wang, J.; Globally asymptotic stability of a kind of third order delay differential system. International Journal of Nonlinear Science 10, No. 1, (2010), 82-87
  45. Yoshizawa, T.; Stability theory by Liapunov's second method, The Mathematical Society of Japan (1966)
  46. Yoshizawa, T.; Stability theory and existence of periodic solutions and almost periodic solutions. Spriger-Verlag, New York. Heidelberg. Berlin (1975)
  47. Zhu, Y. F.; On stability, boundedness and existence of periodic solution of a kind of third-order nonlinear delay differential system. Ann. Differential Equation 8 (2) (1992), 249-259

Полный текст (pdf)