In this paper, we study in Banach spaces the existence of fixed points of asymptotically regular mapping T satisfying: for each x, y in the domain and for n=1, 2,…, $$parallelT^nx-T^nyparallelleq$leq$a_nparallelx-yparallel+b_n (parallelx-T^nxparallel+parallely-T^nyparallely)$$</TEX> where $a_n,; b_n,; C_n$ are nonnegative constants satisfying certain conditions. We also establish some fixed point theorems for these mappings in a Hibert space, in L(sup)p spaces, in Hardy space H(sup)p, and in Soboleve space $H^{k,p} for 1< ho