  In this paper we study the nonlinear Neumann boundary value problem of the
following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla
u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla
u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=\lambda f(x,u) in a bounded smooth
domain $\Omega\subset\mathbb{R}^{N}$ with Neumann boundary condition given by
|\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partial\nu}+|\nabla
u|^{p_{2}(x)-2}\frac{\partial u}{\partial\nu}=\mu g(x,u) on $\partial\Omega$.
Under appropriate conditions on the source and boundary nonlinearities, we
obtain a number of results on existence and multiplicity of solutions by
variational methods in the framework of variable exponent Lebesgue and Sobolev
spaces.
