Strong convergence of averaged iteration for asymptotically non-expansive mappings

Let E be a uniformly convex Banach space E   with a weakly continuous duality mapping JφJφ and K be a non-empty bounded closed convex subset of E  . For an asymptotically non-expansive mapping T:K→KT:K→K, an arbitrary initial values z1,x1∈Kz1,x1∈K and an anchor point u∈Ku∈K, we define iteratively the sequences {zm}{zm} and {xn}{xn} as follows:zm=tmu+(1-tm)1m+1∑j=0mTjzm,m⩾0,xn+1=αnu+(1-αn)1n+1∑j=0nTjxn,n⩾0,where {tm}⊂(0,1){tm}⊂(0,1) and {αn}⊂(0,1){αn}⊂(0,1) satisfy proper conditions. We prove that {zm}{zm} and {xn}{xn} converge strongly to some p∈F(T)p∈F(T), respectively.