General iterative scheme for finding attractive points of （α，β）—mildly hybrid set—valued mappings in Banach spaces

Wang Qianwen　Wei Qiuhong　Zeng Liuchuan

Abstract： In this paper，we first introduce the concepts of （α，β）-mildly hybrid set-valued mappings and condition I*，and then propose a general iterative method for finding attractive points of （α，β） -mildly hybrid set-valued mappings.Moreover，we establish strong convergence of the proposed iterative scheme for （α，β）-mildly hybrid set-valued mappings in a uniformly convex Banach space by using condition I* and demi-compact property.Our results improve and modify the corresponding ones announced by other authors in the literature.

Key words： mildly hybrid set-valued mapping； condition I*； strongly attractive point； attractive point； uniformly convex Banach space

CLC number： O 177.91 Document code： A Article ID： 1000-5137（2018）04-0397-09

摘 要： 引入（α，β）-适度混合集值映像与条件I*的概念，提出了用于寻求（α，β）-适度混合集值映像吸引点的一般迭代法.在一致凸Banach空间中，利用条件I*与半紧性质，建立了所提（α，β）-适度混合集值映像的迭代格式的强收敛性.结论改进与修正了之前文献中已发布的相应结果.

关键词： 适度混合集值映像； 条件I*； 强吸引点； 吸引点； 一致凸Banach空间

References：

[1] Chen L L.A new iterative scheme for finding attractive points of （α，β）-generalized hybrid set-valued mappings [J].Journal of Nonlinear Sciences and Applications，2017，10：1228-1237.

[2] García-Falset J，Llorens-Fuster E，Suzuki T.Fixed point theory for a class of generalized nonexpansive mapping [J].Journal of Mathematical Analysis and Applications，2011，375：185-195.

[3] Kocourek P，Takahashi W，Yao G C.Fixed point theorems and weak convergences theorems for generalized hybrid mappings in Hilbert spaces [J].Taiwanese Journal of Mathematics，2010，14：2497-2511.

[4] Takahashi W.Fixed point theorems for new nonlinear mappings in a Hilbert space [J].Journal of Nonlinear and Convex Analysis，2010，11：79-88.

[5] Zheng Y Z.Attractive points and convergence theorems of generalized hybrid mapping [J].Journal of Nonlinear Sciences and Applications，2015，8：354-362.

[6] Senter H F，Dotson W G.Approximating fixed point of nonexpansive mappings [J].Proceedings of the American Mathematical Society，1974，44：375-380.

[7] Clarkson J A.Uniformly convex space [J].Transactions of the American Mathematical Society，1936，40：396-414.

[8] Xu H K.Inequalities in Banach space with applications [J].Nonlinear Analysis，1991，16：1127-1138.

（責任编辑：冯珍珍）