集值非扩张映象的迭代收敛性及不动点

INTERATIVE CONVERGENCE AND FIXED POINTS OF SET VALUED NONEXPANSIVE MAPPINGS

设Ｄ是赋范空间Ｘ的有界凸子集，Ｔ∶Ｄ→ＣＢ（Ｄ）是δ集值非扩张映象．给定Ｄ中序列ｘｎ｛｝和两个实数列ｔｎ｛｝和ｓｎ｛｝，满足（ｉ）０≤ｔｎ≤ｔ＜１和∞ｎ＝１ｔｎ＝∞，（ｉ）０≤ｓｎ≤１，∞ｎ＝１Ｓｎ＜∞和ｌｉｍｎ→∞ｔｎ－１ｓｎ＝０，（ｉｉ）ｘｎ＋１∈ｔｎＴｙｎ＋（１＋ｔｎ）ｘｎ，ｙｎ∈ｓｎＴｘｎ＋（１－ｓｎ）ｘｎ，ｎ＝１，２，３…．则ｌｉｍｎ→∞ｄ（ｘｎ，Ｔｘｎ）＝０．并在一定条件下证明了集值映象Ｔ存在不动点以及确保迭代过程收敛到不动点的条件．

Let D be a bounded concex subset of a normed space X and T∶D→CB(D) be a δ set valued nonexpansive mapping.Given a sequencex n in D and two real numbers sequences x nand s n. satisfyin (i) 0≤t n≤t<1 and ∞n=1t n=∞,(ii )0≤s n≤1,∞n=1s n<∞ and linn=∞t n -1 s n=0,(iii)x n+1 ∈t nTy n+(1+t n)x n,y n∈s nTx n+(1-s n)x n,n=1,2,3….Then limn=∞d(x n,Tx n)=0.Presented in the paper are the existence of fixed points for set valued mapping T and requirement ensuring convergence to fixed points in interative process.