General Regularization Methods For Nonexpansive Mappings In Banach Spaces

With the rapid development of mathematics and computer science,computer tools obtain great progress,which makes large scale scientific and engineering computing possible.The research about the fixed point iteration algorithm of nonlinear operator and variational inequalities in the Hilbert space is flourish and the results is very fruitful which is widely used to control theory,game theory,the theory of economic equilibrium, social and economic model nonlinear programming,transportation and engineering. Therefore,the study of fixed point method has theoretical and practical significance. However,most scholars research the iterative algorithm in the Hilbert space. The research in the Banach spaces is still relatively small.We study the strong convergence of nonexpansive mappings iterative algorithm in Banach spaces in this paper.Let X be a uniformly smooth Banach space and C a closed convex subset of X. Consider a nonexpansive mapping Tand assume the set Fix(T) of fixed points of T is nonempty. The iterative algorithm of regularization method in Hilbert space and the basic conclusion of the consistent smooth banach space are used in the paper. Firstly,Consider The general regularization method:xn+1=T(αnf(xn)+(1-αn)xn n≥1,Beginning with an arbitrary initial guess x0,while{αn)(?)(0,1). Assume (i)αn→0(n→∞)；(ii)∑n=0∞ αn=∞;(iii)∑n=1∞|αn+1-αn|<∞ or lim αn/αn+1=1, Then the sequence (x)converges in norm t0 {xn)→Q(f),where Q:C→Fix(T) is the sunny nonexpansive.Then,by changing the iteration formula of the algorithm: yb=T(αnf(xn)+(1-αn)xn) xn+1=λxn+(1-λ)yn Assume(i)αn→0(n→∞)、(ii)∑n=0∞ αn=∞下，依然证明了序列{xn}→Q(f),or Q:C→Fix(T)is the sunny nonexpansive.This algorithm reduces constraints,thus the scope of use more widely.