Local negative circuits and fixed points in Boolean networks

To each Boolean function F from {0,1}^n to itself and each point x in {0,1}^n, we associate the signed directed graph G_F(x) of order n that contains a positive (resp. negative) arc from j to i if the partial derivative of f_i with respect of x_j is positive (resp. negative) at point x. We then focus on the following open problem: Is the absence of a negative circuit in G_F(x) for all x in {0,1}^n a sufficient condition for F to have at least one fixed point? As main result, we settle this problem under the additional condition that, for all x in {0,1}^n, the out-degree of each vertex of G_F(x) is at most one.