M. B. de Souza Cortes
Departamento de Estatistica - Universidade Estadual de Maringa
87020-900 Maringa - PR, Brazil
E-mail matofu@uem.br
Abstract.
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.