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,⋯,‖Tnx−Tny‖≤an‖x−y‖+bn(‖x−Tnx‖+‖y−Tny‖)+cn(‖x−Tny‖+‖y−Tny‖),
where an,bn,cn are nonnegative constants satisfying certain conditions. This result generalizes a fixed
point theorem of Górnicki [1].