Summary: The well-known Schauder result on the existence of \(\text{Lip}_\alpha (\overline{\Omega})\) solutions of the Dirichlet problem for bounded domains with smooth boundaries is true for the Helmholtz equation \(\Delta u+\lambda u = 0\) for \(\lambda \leq 0\). We suggest a method of constructing the solution based on an averaging procedure and mean-value theorem. We show some conditions under which, for \(0 < \alpha < 1\), and \(\lambda\leq 0\), a sequence of iterated averages of an initial approximation converges geometrically to the solution.