  The goal of this article is to initiate the study of estimates of the
non-classical structure in the generalized Schottky group theory. In fact, in
this paper, we have investigated the non-classical Schottky structure on a type
of generalized Schottky group (for instance Fuchsian Schottky groups) in the
hyperbolic plane with boundary. A Schottky group is said to be non-classical if
the curves used in the Schottky construction are Jordan curves. More precisely,
in this manuscript, we have provided a structure of the rank $2$ non-classical
Fuchsian Schottky groups by utilizing two suitable hyperbolic M\"obius
transformations on the upper-half plane model. In particular, we have derived
two non-trivial examples of non-classical Fuchsian Schottky groups in the
upper-half plane with the circle at infinity as the boundary.
