A nonlinear differential equation involving reflection of the argument

T. F. Ma,   E. S. Miranda and M. B. de Souza Cortes

 T. F. Ma and  E. S. Miranda
 Departamento de Matematica - Universidade Estadual de Maringa
 87020-900 Maringa - PR, Brazil

M. B. de Souza Cortes
Departamento de Estatistica - Universidade Estadual de Maringa
87020-900 Maringa - PR, Brazil

E-mail  matofu@uem.br

We study the nonlinear boundary value problem involving reflection of
the argument
-M\Big(\int_{-1}^1\vert u'(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.

AMSclassification. 34B15.

Keywords. Reflection, Brouwer fixed point, Kirchhoff equation.