We study a nonlinear Steklov problem involving weighted p(.)-Laplacian-like operator. Using some variational methods, we obtain the existence and multiplicity of solutions for the following problem \(\begin{aligned} \left\{ \begin{array}{cc} \text {div}\left( \left( 1+\frac{\left| \nabla u\right| ^{p(x)}}{ \sqrt{1+\left| \nabla u\right| ^{2p(x)}}}\right) a(x)\left| \nabla u\right| ^{p(x)-2}\nabla u\right) =b(x)\left| u\right| ^{p(x)-2}u, & \text {in }\Omega \\ \left( 1+\frac{\left| \nabla u\right| ^{p(x)}}{\sqrt{1+\left| \nabla u\right| ^{2p(x)}}}\right) a(x)\left| \nabla u\right| ^{p(x)-2}\frac{\partial u}{\partial \upsilon }=\lambda f(x,u), & \text {on } \partial \Omega ,\end{array} \right. \end{aligned}\) under some suitable conditions.
