Geodesic
Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 43 (2004) no. 1, pp. 7-20 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 34C25, 34D20, 34D40
Keywords: ultimate boundedness; complete Lyapunov functions; nonlinear third order system 
@article{AUPO_2004_43_1_a0,
     author = {Afuwape, A.~U. and Omeike, M. O.},
     title = {Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {7--20},
     year = {2004},
     volume = {43},
     number = {1},
     mrnumber = {2124598},
     zbl = {1068.34052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2004_43_1_a0/}
}
TY  - JOUR
AU  - Afuwape, A. U.
AU  - Omeike, M. O.
TI  - Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2004
SP  - 7
EP  - 20
VL  - 43
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2004_43_1_a0/
LA  - en
ID  - AUPO_2004_43_1_a0
ER  - 
%0 Journal Article
%A Afuwape, A. U.
%A Omeike, M. O.
%T Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2004
%P 7-20
%V 43
%N 1
%U http://geodesic.mathdoc.fr/item/AUPO_2004_43_1_a0/
%G en
%F AUPO_2004_43_1_a0
Afuwape, A. U.; Omeike, M. O. Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 43 (2004) no. 1, pp. 7-20. http://geodesic.mathdoc.fr/item/AUPO_2004_43_1_a0/

[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

[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. | MR | Zbl

[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. | MR | Zbl

[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

[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

[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. | MR | Zbl

[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

[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. | MR | Zbl

[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

[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

[11] Reissig R., Sansone G., Conti R.: Non-Linear Differential Equations of Higher Order. : Noordhoff, Groningen. 1974. | MR

[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