# A nonlinear differential equation involving reflection of the argument

To Fu Ma; E. S. Miranda; M. B. de Souza Cortes

Archivum Mathematicum (2004)

- Volume: 040, Issue: 1, page 63-68
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topMa, To Fu, Miranda, E. S., and de Souza Cortes, M. B.. "A nonlinear differential equation involving reflection of the argument." Archivum Mathematicum 040.1 (2004): 63-68. <http://eudml.org/doc/249316>.

@article{Ma2004,

abstract = {We study the nonlinear boundary value problem involving reflection of the argument \[ -M\Big (\int \_\{-1\}^1\vert u^\{\prime \}(s)\vert ^2\,ds\Big )\,u^\{\prime \prime \}(x) = f\big (x,u(x),u(-x)\big ) \quad \quad x \in [-1,1]\,, \]
where $M$ and $f$ are continuous functions with $M>0$. Using Galerkin approximations combined with the Brouwer’s fixed point theorem we obtain existence and uniqueness results. A numerical algorithm is also presented.},

author = {Ma, To Fu, Miranda, E. S., de Souza Cortes, M. B.},

journal = {Archivum Mathematicum},

keywords = {reflection; Brouwer fixed point; Kirchhoff equation; Brouwer fixed point; Kirchhoff equation},

language = {eng},

number = {1},

pages = {63-68},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {A nonlinear differential equation involving reflection of the argument},

url = {http://eudml.org/doc/249316},

volume = {040},

year = {2004},

}

TY - JOUR

AU - Ma, To Fu

AU - Miranda, E. S.

AU - de Souza Cortes, M. B.

TI - A nonlinear differential equation involving reflection of the argument

JO - Archivum Mathematicum

PY - 2004

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 040

IS - 1

SP - 63

EP - 68

AB - We study the nonlinear boundary value problem involving reflection of the argument \[ -M\Big (\int _{-1}^1\vert u^{\prime }(s)\vert ^2\,ds\Big )\,u^{\prime \prime }(x) = f\big (x,u(x),u(-x)\big ) \quad \quad x \in [-1,1]\,, \]
where $M$ and $f$ are continuous functions with $M>0$. Using Galerkin approximations combined with the Brouwer’s fixed point theorem we obtain existence and uniqueness results. A numerical algorithm is also presented.

LA - eng

KW - reflection; Brouwer fixed point; Kirchhoff equation; Brouwer fixed point; Kirchhoff equation

UR - http://eudml.org/doc/249316

ER -

## References

top- Arosio A., Panizzi S., On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330. (1996) Zbl0858.35083MR1333386
- Chipot M., Rodrigues J. F., On a class of nonlinear nonlocal elliptic problems, RAIRO Modél. Math. Anal. Numér. 26 (1992), 447–467. (1992) MR1160135
- Gupta C. P., Existence and uniqueness theorems for boundary value problems involving reflection of the argument, Nonlinear Anal. 11 (1987), 1075–1083. (1987) Zbl0632.34069MR0907824
- Hai D. D., Two point boundary value problem for differential equations with reflection of argument, J. Math. Anal. Appl. 144 (1989), 313–321. (1989) Zbl0699.34017MR1027038
- Kesavan S., Topics in Functional Analysis and Applications, Wiley Eastern, New Delhi, 1989. (1989) Zbl0666.46001MR0990018
- Ma T. F., Existence results for a model of nonlinear beam on elastic bearings, Appl. Math. Lett. 13 (2000), 11–15. Zbl0965.74030MR1760256
- O’Regan D., Existence results for differential equations with reflection of the argument, J. Austral. Math. Soc. Ser. A 57 (1994), 237–260. (1994) Zbl0818.34037MR1288675
- Sharma R. K., Iterative solutions to boundary-value differential equations involving reflection of the argument, J. Comput. Appl. Math. 24 (1988), 319–326. (1988) Zbl0664.65080MR0974020
- Wiener J., Aftabizadeh A. R., Boundary value problems for differential equations with reflection of the argument, Int. J. Math. Math. Sci. 8 (1985), 151–163. (1985) Zbl0583.34055MR0786960

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.