Iterative Algorithm And Application For Approximating Fixed Points Of General Nonexpansive Mappings

In this paper, firstly for a uniformly convex Banach space E whose norm is uniformly G(a|^)teaux differentiable and C a nonempty closed convex subset of E, let {T_n} be a sequence of nonexpansive mappings of C into itself. By using iterative method, we prove the sequence {x_n} converges strongly to Q_x, where Q is the sunny nonexpansive retraction of E onto F(T)=∩_n=1^∞F(T_n). Our results extend and improve the results of Koji Aoyamaa and Yasunori Kimurab and some others.Secondly, in a Hilbert space H by combining the qusi-nonexpansive mapping of C into itself and the monotone hybrid method we get new iterative sequence {x_n}. By using the technique of monotone hybrid algorithm, we prove that the sequence {x_n} converges strongly to the point P_F(T)x₀, where P_F(T) is the metric projection from C to F(T). Then under the condition of monotone hybrid algorithm,we consider the relative nonexpansive mapping in uniformly smooth Banach space and obtain strong convergence results of the sequence {x_n}.This conclusion modify and improve the relative results of S.Matsushita and W.Takahashi, Nakajo, Takahashi, Kim, and so on.Finally we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. At the beginning, we approximate the common element of the set of the solutions through the viscosity approximation method and then by the results we have got a optimization problem is considered. At last we obtain the optimization approximating element of the iterative algorithm. Thus the results of Combettes and Hirstoaga, Moudafi, TAda and Takahashi etc. were improved.