  A new generalization of the Browder's degree for the mappings of the type
$(S)_+$ is presented. The main idea is rooted in the observation that the
Browder's degree remains unchanged for the mappings of the form $A: Y\to X^*$,
where $Y$ is a reflexive uniformly convex Banach space continuously embedded in
the Banach space $X$. The advantage of the suggested degree lies in the
simplicity it provides for the calculations of the degree associated with
nonlinear operators. An application from the theory of phase transition in
liquid crystals is presented for which the suggested degree has been
successfully applied.
