  By applying Ricceri's variational principle, we demonstrate the existence of
solutions for the following Robin problem
  \begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( \omega
_{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla u\right) =\lambda \omega
_{2}(x)f(x,u), & x\in \Omega \\ \omega _{1}(x)\left\vert \nabla u\right\vert
^{p(x)-2}\frac{\partial u}{ \partial \upsilon }+\beta (x)\left\vert
u\right\vert ^{p(x)-2}u=0, & x\in \partial \Omega , \end{array} \right.
\end{equation*}
  in $W_{\omega _{1},\omega _{2}}^{1,p(.)}\left( \Omega \right) $ under some
appropriate conditions.
