In the present paper, we first give some examples of self-mappings which are asymptoticaly nonexpansive in the intermediate, not strictly hemicontractive, but satisfy the property (H). It is then shown that the modified Mann and Ishikawa iteration processes defined by $x_{n+1}=(1-alpha_n)x_n+alpha_nT^nx_n and x_{n+1}=(1-alpha_n)x_n+alpha_nT^n[(1-eta_n)x_n+eta_nT^nx_n]$,respectively, converges strongly to the unique fixed point of such a self-mapping in general Banach spaces.