| Title:
|
Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations (English) |
| Author:
|
Afuwape, A. U. |
| Author:
|
Omeike, M. O. |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
43 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
7-20 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $${\ensuremath{\mathop{\smash{X}\vrule width0ptheight5.46pt}\limits^{\hbox to 8pt{\hss \footnotesize \kern1pt.\kern-0.065em.\kern-0.065em.\hss}}}}+F(\ddot{X})+G(\dot{X})+H(X)= P(t,X,\dot{X},\ddot{X})$$ where $X,F(\ddot{X})$, $G(\dot{X})$, $H(X)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $P:\mathbb{R}\times \mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ continuous in their respective arguments. We do not necessarily require that $F(\ddot{X}),G(\dot{X})$ and $H(X)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized. (English) |
| Keyword:
|
ultimate boundedness |
| Keyword:
|
complete Lyapunov functions |
| Keyword:
|
nonlinear third order system |
| MSC:
|
34C25 |
| MSC:
|
34D20 |
| MSC:
|
34D40 |
| idZBL:
|
Zbl 1068.34052 |
| idMR:
|
MR2124598 |
| . |
| Date available:
|
2009-08-21T12:53:36Z |
| Last updated:
|
2012-05-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/132943 |
| . |
| Reference:
|
[1] Afuwape A. U.: Ultimate boundedness results for a certain system of third-order non-linear differential equation.J. Math. Anal. Appl. 97 (1983), 140–150. MR 0721235 |
| Reference:
|
[2] Afuwape A. U.: Uniform dissipative solutions for a third-order non-linear differential equation.In: Differential equations (J. W. Knowles and R. T. Lewis, eds.), Elsevier, North Holland, 1984, 1–6. Zbl 0552.34060, MR 0799326 |
| Reference:
|
[3] Afuwape A. U.: Further ultimate boundedness results for a third order non-linear system of differential equations.Analisi Funzionale e Appl. 6, 99–100, N. I. (1985), 348–360. Zbl 0592.34024, MR 0805225 |
| Reference:
|
[4] Afuwape A. U., Ukpera A. S.: Existence of solutions of periodic boundary value problems for some vector third order differential equations.J. of Nig. Math. Soc. 20 (2001), 1–17. MR 2055195 |
| Reference:
|
[5] Ezeilo J. O. C.: $n$-dimensioinal extensions of boundedness and stability theorems for some third order differential equations.J. Math. Anal. Appl. 18 (1967), 395–416. MR 0212298 |
| Reference:
|
[6] Ezeilo J. O. C.: Stability Results for the Solutions of some third and fourth order differential equations.Ann. Mat. Pura. Appl. 66, 4 (1964), 233–250. Zbl 0126.30403, MR 0173831 |
| Reference:
|
[7] Ezeilo J. O. C.: New properties of the equation $x^{\prime \prime \prime }+ax^{\prime \prime }+bx^{\prime }+h(x)=p(t,x,\dot{x},x^{\prime \prime })$ for certain special values of the incrementary ratio $y^{-1}\lbrace h(x+y)-h(x)\rbrace $.In: Equations Differentielles et Functionelles Non-lineares (P. Janssens, J. Mawhin and N. Rouche, eds.), Hermann, Paris, 1973, 447–462. MR 0430413 |
| Reference:
|
[8] Ezeilo J. O. C., Tejumola H. O.: Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations.Ann. Math. Pura e Appl. 74 (1966), 283–316. Zbl 0148.06701, MR 0204787 |
| Reference:
|
[9] Ezeilo J. O. C., Tejumola H. O.: Further results for a system of third order ordinary differential equations.Atti. Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 143–151. MR 0425261 |
| Reference:
|
[10] Meng F. W.: Ultimate boundedness results for a certain system of third order nonlinear differential equations.J. Math. Anal. Appl. 177 (1993), 496–509. MR 1231497 |
| Reference:
|
[11] Reissig R., Sansone G., Conti R.: Non-Linear Differential Equations of Higher Order. : Noordhoff, Groningen., 1974. MR 0344556 |
| Reference:
|
[12] Tiryaki A.: Boundedness and periodicity results for a certain system of third order non-linear differential equations.Indian J. Pure Appl. Math. 30, 4 (1999). 361–372. MR 1695688 |
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