On a first-order accurate difference scheme for a singularly perturbed problem with a turning point

A singularly perturbed problem with a turning point is considered. The solution has a boundary layer of exponential type in a neighborhood of a boundary point. The problem is solved approximately by means of a difference scheme of exponential fitting on a uniform grid. It is proved that the solutions obtained from this scheme converge uniformly with respect to the perturbation parameter with the first order of accuracy to the solution of the original differential problem as the grid step tends to zero.