Fixed point theorems for generalized Lipschitzian semigroups are proved in p -uniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in L p spaces, in Hardy space H p , and in Sobolev spaces H k , p , for 1 < p < ∞ and k ≥ 0 .