Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points are possibly on the boundary. In reading some textbook, I find that the fixed points on the boundary can not be involved in the fixed-point index. I wonder whether there is an index theorem that treats the fixed points on the boundary?