Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations

A. U. Afuwape; Mathew Omonigho Omeike

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2004)

  • Volume: 43, Issue: 1, page 7-20
  • ISSN: 0231-9721

Abstract

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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 X w i d t h 0 p t h e i g h t 5 . 46 p t t o 8 p t . . . + F ( X ¨ ) + G ( X ˙ ) + H ( X ) = P ( t , X , X ˙ , X ¨ ) where X , F ( X ¨ ) , G ( X ˙ ) , H ( X ) , P ( t , X , X ˙ , X ¨ ) are real n -vectors with F , G , H : ℝ n → ℝ n and P : ℝ × ℝ n × ℝ n × ℝ n → ℝ n continuous in their respective arguments. We do not necessarily require that F ( X ¨ ) , G ( X ˙ ) and H ( X ) are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.

How to cite

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Afuwape, A. U., and Omeike, Mathew Omonigho. "Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 43.1 (2004): 7-20. <http://eudml.org/doc/32353>.

@article{Afuwape2004,
abstract = {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 \[\{\mathop \{\smash\{X\}\vrule width0ptheight5.46pt\}\limits ^\{\hbox\{t\}o 8pt\{\hss \footnotesize \hspace\{1.0pt\}.\hspace\{-0.65002pt\}.\hspace\{-0.65002pt\}.\hss \}\}\}+F(\ddot\{X\})+G(\dot\{X\})+H(X)= P(t,X,\dot\{X\},\ddot\{X\})\] 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.},
author = {Afuwape, A. U., Omeike, Mathew Omonigho},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {ultimate boundedness; complete Lyapunov functions; nonlinear third order system; ultimate boundedness; complete Lyapunov functions; nonlinear third-order system},
language = {eng},
number = {1},
pages = {7-20},
publisher = {Palacký University Olomouc},
title = {Further ultimate boundedness of solutions of some system of third order nonlinear ordinary differential equations},
url = {http://eudml.org/doc/32353},
volume = {43},
year = {2004},
}

TY - JOUR
AU - Afuwape, A. U.
AU - Omeike, Mathew Omonigho
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
PB - Palacký University Olomouc
VL - 43
IS - 1
SP - 7
EP - 20
AB - 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 \[{\mathop {\smash{X}\vrule width0ptheight5.46pt}\limits ^{\hbox{t}o 8pt{\hss \footnotesize \hspace{1.0pt}.\hspace{-0.65002pt}.\hspace{-0.65002pt}.\hss }}}+F(\ddot{X})+G(\dot{X})+H(X)= P(t,X,\dot{X},\ddot{X})\] 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.
LA - eng
KW - ultimate boundedness; complete Lyapunov functions; nonlinear third order system; ultimate boundedness; complete Lyapunov functions; nonlinear third-order system
UR - http://eudml.org/doc/32353
ER -

References

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  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. (1984) Zbl0552.34060MR0799326
  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. (1985) Zbl0592.34024MR0805225
  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. MR2055195
  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. (1967) MR0212298
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  7. Ezeilo J. O. C., New properties of the equation x ' ' ' + a x ' ' + b x ' + h ( x ) = p ( t , x , x ˙ , x ' ' ) for certain special values of the incrementary ratio y - 1 { h ( x + y ) - h ( x ) } , In: Equations Differentielles et Functionelles Non-lineares (P. Janssens, J. Mawhin and N. Rouche, eds.), Hermann, Paris, 1973, 447–462. (1973) MR0430413
  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. (1966) Zbl0148.06701MR0204787
  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. (1975) MR0425261
  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. (1993) MR1231497
  11. Reissig R., Sansone G., Conti R.: Non-Linear Differential Equations of Higher Order., Noordhoff, Groningen, , 1974. (1974) MR0344556
  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. (1999) MR1695688

Citations in EuDML Documents

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  1. Irena Rachůnková, Jan Tomeček, Singular nonlinear problem for ordinary differential equation of the second order
  2. Mathew Omonigho Omeike, Stability and Boundedness of Solutions of a Certain System of Third-order Nonlinear Delay Differential Equations
  3. Mathew Omonigho Omeike, A. U. Afuwape, New result on the ultimate boundedness of solutions of certain third-order vector differential equations
  4. Mathew Omonigho Omeike, O. O. Oyetunde, A. L. Olutimo, Boundedness of Solutions of Certain System of Second-order Ordinary Differential Equations
  5. Larbi Fatmi, Moussadek Remili, Stability and Boundedness of Solutions of Some Third-order Nonlinear Vector Delay Differential Equation
  6. Djamila Beldjerd, Moussadek Remili, Boundedness and square integrability of solutions of certain third-order differential equations
  7. Anthony Uyi Afuwape, Mathew Omonigho Omeike, Ultimate boundedness of some third order ordinary differential equations

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