Strong convergence theorems on two iterative method for non-expansive mappings

In this paper, we study the convergence of two type iteration processes as follows:(I)xn+1=αnx+(1-αn)T(βnx+(1-βn)Txn),(II)xn+1=αn(βnx+(1-βn)Txn)+(1-αn)Anxn,where An=1n+1∑j=0nTj:C→C in uniformly convex Banach space X, which possesses a weakly sequentially continuous duality mapping J and in uniformly convex Banach space X with a uniformly Gateaux differentiable norm, respectively. And prove that above sequences converge strongly to Px when the real sequence {αn}, {βn} satisfies appropriate conditions, where P is sunny non-expansive from C onto F(T).