Geodesic
A nonlinear differential equation involving reflection of the argument
Archivum mathematicum, Tome 40 (2004) no. 1, pp. 63-68 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 34B15
Keywords: reflection; Brouwer fixed point; Kirchhoff equation 
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Ma, T. F.; Miranda, E. S.; de Souza Cortes, M. B. A nonlinear differential equation involving reflection of the argument. Archivum mathematicum, Tome 40 (2004) no. 1, pp. 63-68. http://geodesic.mathdoc.fr/item/ARM_2004_40_1_a6/

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