A new construction of the degree of maximal monotone maps

The inclusion equations of the type $f\in T(x)$ where $T: X \to 2^{X^{\ast}}$ is a maximal monotone map, are extensively studied in nonlinear analysis. In this paper, we present a new construction of the degree of maximal monotone maps of the form $T:Y\to 2^{X^{\ast }}$, where $Y$ is a locally uniformly convex and separable Banach space continuously embedded in the uniformly convex Banach space $X$. The advantage of the new construction lies in the remarkable simplicity it offers for calculation of degree in comparison with the classical one suggested by F. Browder. We prove a few classical theorems in convex analysis through the suggested degree.