Summary: In this article, we consider a Neumann boundary value problem driven by \(p (x)\)-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: \[ \begin{aligned} \begin{cases} - \Delta_{p (x)}^l u + \delta | u |^{\zeta (x) - 2} u = \mu g (x, u) + \lambda f (x, u, \nabla_u) \quad &\text{ in } \Omega,\\ \frac{\partial_u}{\partial \eta} = 0 \quad &\text{ on } \partial \Omega, \end{cases} \end{aligned} \] where \(\Delta_{p (x)}^l u\) is the \(p (x)\)-Laplacian-like operator, \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(\delta\), \(\mu\) and \(\lambda\) are three real parameters, \(p (x)\), \(\zeta (x) \in C_+ (\overline{\Omega})\), \(\eta\) is the outer unit normal to \(\partial \Omega\) and \(g\), \(f\) are Carathéodory functions. Under suitable nonstandard growth conditions on \(g\) and \(f\) and using the topological degree for a class of demicontinuous operator of generalized \((S_+)\) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.