Singular nonlinear problem for ordinary differential equation of the second order

Irena Rachůnková; Jan Tomeček

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2007)

  • Volume: 46, Issue: 1, page 75-84
  • ISSN: 0231-9721

Abstract

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The paper deals with the singular nonlinear problem u ' ' ( t ) + f ( t , u ( t ) , u ' ( t ) ) = 0 , u ( 0 ) = 0 , u ' ( T ) = ψ ( u ( T ) ) , where f ∈ 𝐶𝑎𝑟 ( ( 0 , T ) × D ) , D = ( 0 , ∞ ) × . We prove the existence of a solution to this problem which is positive on ( 0 , T ] under the assumption that the function f ( t , x , y ) is nonnegative and can have time singularities at t = 0 , t = T and space singularity at x = 0 . The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

How to cite

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Rachůnková, Irena, and Tomeček, Jan. "Singular nonlinear problem for ordinary differential equation of the second order." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 46.1 (2007): 75-84. <http://eudml.org/doc/32463>.

@article{Rachůnková2007,
abstract = {The paper deals with the singular nonlinear problem \[ u^\{\prime \prime \}(t) + f(t,u(t),u^\{\prime \}(t)) = 0,\quad u(0) = 0,\quad u^\{\prime \}(T) = \psi (u(T)), \] where $f \in \mathop \{\mathit \{Car\}\}((0,T)\times D)$, $D = (0,\infty )\times $. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t = 0$, $t = T$ and space singularity at $x = 0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.},
author = {Rachůnková, Irena, Tomeček, Jan},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity; singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity},
language = {eng},
number = {1},
pages = {75-84},
publisher = {Palacký University Olomouc},
title = {Singular nonlinear problem for ordinary differential equation of the second order},
url = {http://eudml.org/doc/32463},
volume = {46},
year = {2007},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Tomeček, Jan
TI - Singular nonlinear problem for ordinary differential equation of the second order
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2007
PB - Palacký University Olomouc
VL - 46
IS - 1
SP - 75
EP - 84
AB - The paper deals with the singular nonlinear problem \[ u^{\prime \prime }(t) + f(t,u(t),u^{\prime }(t)) = 0,\quad u(0) = 0,\quad u^{\prime }(T) = \psi (u(T)), \] where $f \in \mathop {\mathit {Car}}((0,T)\times D)$, $D = (0,\infty )\times $. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t = 0$, $t = T$ and space singularity at $x = 0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.
LA - eng
KW - singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity; singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity
UR - http://eudml.org/doc/32463
ER -

References

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  5. 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
  6. Omeike M. O., Qualitative Study of solutions of certain n-system of third order non-linear ordinary differential equations, Ph.D. Thesis, University of Agriculture, Abeokuta, 2005. 
  7. Reissig R., Sansone G., Conti R.: Non-linear Differential Equations of higher order., No-ordhoff International Publishing, , 1974. (1974) MR0344556
  8. Tejumola H. O., On a Lienard type matrix differential equation, Atti Accad. Naz. Lincei Rendi. Cl. Sci. Fis. Mat. Natur (8) 60, 2 (1976), 100–107. (1976) Zbl0374.34035MR0473341
  9. Yoshizawa T.: Stability Theory by Liapunov’s Second Method., The Mathematical Society of Japan, , 1966. (1966) MR0208086

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