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We study the nonlinear boundary value problem involving reflection of the argument $$-M\left({\int }_{-1}^1{\left|u^{\text{'}}\left(s\right)\right|}^{2}ds\right)u^{\text{'}\text{'}}\left(x\right)=f(x,u\left(x\right),u(-x))x\in [-1,1],$$ 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.