Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces ✩
Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces ✩
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DOI：
10.1016/j.jmaa.2008.01.045
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发表时间：
2008-07
影响因子：
1.3
通讯作者：
Haiyun Zhou
中科院分区：
文献类型：
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作者：
Haiyun Zhou
让C是一个闭凸子集的一个真正的希尔伯特空间H和假设T是一个κ严格pseudo-contraction C .考虑曼的迭代算法给出的证明,如果选择的控制序列{αn}所以κ<αn < 1和∑n = 0∞(αn−κ)(1−αn) =∞,然后描写→∞为xn−时候为= d (0, R (a)¯),我在哪里a =−T和d (0, d)表示之间的距离的起源和子集组d H .由于这一结果,我们证明如果T有定点在C语言中,则{xn}弱收敛于t的不动点。同时，我们将Reich的结果推广到Hilbert空间下的κ-严格伪收缩。进一步，利用杂化投影，建立了Lipschitz伪收缩的强收敛定理。本文的结果改进或推广了Browder和Petryshyn [F.E.]的相应结果张志强，Hilbert空间中非线性映射不动点的构造，数学学报。分析的。《中国科学》第20卷第1期[B.E.]Rhoades，用无穷矩阵的不动点迭代，译。阿米尔。数学。社会科学（1974）162-176 [j]。马里诺,H.-K。Xu， Hilbert空间中严格伪压缩的弱和强收敛定理，数学。分析的。应用学报，329(1)(2007)336-346]。
Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by It is proved that if the control sequence {αn} is chosen so that κ<αn<1 and ∑n=0∞(αn−κ)(1−αn)=∞, then limn→∞‖xn−Txn‖=d(0,R(A)¯), where A=I−T and d(0,D) denotes the distance between the origin and the subset set D of H. As a consequence of this result, we prove that if T has a fixed point in C, then {xn} converges weakly to a fixed point of T. Also, we extend a result due to Reich to κ-strict pseudo-contractions in the Hilbert space setting. Further, by virtue of hybridization projections, we establish a strong convergence theorem for Lipschitz pseudo-contractions. The results presented in this paper improve or extend the corresponding results of Browder and Petryshyn [F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197–228], Rhoades [B.E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 (1974) 162–176] and of Marino and Xu [G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (1) (2007) 336–346].