Abstract: As a generalization of Fan-Glicksberg fixed point theorem, a new existence theorem of fixed point theorem for set-valued mapping is proved. And then, by introducing the concept of weakly quasiconcave function, and replacing quasiconcave function by weakly quasiconcave function, a new existence theorem of Nash equilibrium point for game theory is deduced. Lastly an example is given to prove that the new theory is more useful.