Strongly Convergence Theorem of m-accretive Operators

Yujun Liu,Sumei Xu

doi:10.5539/mas.v1n2p41

Abstract

Let X be a real Banach space, we will introduce a modifieations of the Mann iterations in a uniformly smooth Banach, Where satisfied some conditions, then we will prove the strongly space, ((1))(1)xuxJx {}, 1nnnnrn n to a zero of accretive operators. This theorem extend (Kim and Xu, Nonlinear convergence of the sequence {}x n Analysis) results. (Kim, 2005, pp.51-60)

Highlights

Since J is uniformly continuous on bounded sets of X, we find that
Let t o 0, we can get lim sup u p, J d 0 nof we claim that xn o p in norm
Strong convergence theorem for infinite families of no expansive mappings in general Banach spaces

Summary

Jn d lim supJ n nof

Let X be a uniformly smooth Banach space, C a closed convex subset of X and T : C o C a nonexpansive mapping such that Fix(T ) z I. For any given sequence {Dn} in [0,1], O (0,1) and u, x0 C , defined {xn} by the iterative algorithm xn 1 D n (O u (1 O ) xn ) (1 D n ) J rn xn ,. Proof: We first show that {xn} is bounded. Take a point p Fix(T ) to get, using the nonexpansive of Jr , xn p d DnO u p Dn (1 O) xn p (1Dn ) Jr xn p d DnO u p Dn (1 O) xn p (1Dn ) xn p d DnO u p (1DnO) xn p. We have xn Jrn xn Dn Ou (1 O ) xn Jrn xn o 0 (n o f)

Dn Jn

Put p