A fixed point theorem is proved in a Banach space E which has uniformly normal structure for asymptotically regular mapping T satisfying: for each x , y in the domain and for n = 1 , 2 , ⋯ , ‖ T n x − T n y ‖ ≤ a n ‖ x − y ‖ + b n ( ‖ x − T n x ‖ + ‖ y − T n y ‖ ) + c n ( ‖ x − T n y ‖ + ‖ y − T n y ‖ ) , where a n , b n , c n are nonnegative constants satisfying certain conditions. This result generalizes a fixed point theorem of Górnicki [1].