Iterative Approximation of Fixed Points for Almost Asymptotically Nonexpansive Type Mappings in Banach Spaces
-
摘要: 在Banach空间中引入了一类新的几乎渐近非扩张型映象,概括了Banach空间中若干熟知的非线性的Lipschitz映象类与非Lipschitz映象类成特例;例如,熟知的非扩张映象类,渐近非扩张映象类与渐近非扩张型映象类.考虑了用于逼近几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的收敛性问题.关于Banach空间范数的S.S.Chang的不等式与H.K.Xu的不等式皆被用于做精确不动点与近似不动点间的误差估计.而且,张石生教授用于做带误差的修改了的Ishikawa迭代序列收敛性分析的方法(应用数学和力学,2001,22(1):23-31)被推广到几乎渐近非扩张型映象的情况.给出了用于求一致凸Banach空间中几乎渐近非扩张型映象不动点的带误差的修改了的Ishikawa迭代序列的新的收敛判据.并且,由该判据,立即得到了此类映象的带误差的修改了的Mann迭代序列的新的收敛判据.上述结果统一、改进与推广了张石生教授关于用带误差的修改了的Ishikawa与Mann迭代序列来逼近渐近非扩张型映象不动点方面的结果.
-
关键词:
- 几乎渐近非扩张型映象 /
- 不动点 /
- 带误差的修改了的Ishikawa迭代序列 /
- 带误差的修改了的Mann迭代序列
Abstract: A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced, which includes a number of known classes of nonlinear Lipschitzian mappings and non-Lipschitzian mappings in Banach spaces as special cases; for example, the known classes of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings. The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered. Not only S. S. Chang's inequality but also H. K. Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one. Moreover, ZHANG Shi-sheng's method(Applied Mathema tics an d Mechanics(English Edition), 2001, 22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also, the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria. The above results unify, improve and generalize ZHANG Shi-sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors. -
[1] Goebel K,Kirk W A.A fixed point theorem for asymptotically nonexpansive mappings[J].Proc AmerMath Soc,1972,35(1):171-174. [2] Kirk W A.Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type[J].Israel J Math,1974,11:339-346. [3] Xu H K.Inequalities in Banach spaces with applications[J].Nonlinear Anal TMA,1991,16(12):1127-1138. [4] 张石生.关于Banach空间中渐近非扩张型映象的不动点的迭代逼近问题[J].应用数学和力学,2001,22(1):25-34. [5] Reich S.Strong convergence theorems for resolvents of accretive operators in Banach spaces[J].J Math AnalAppl,1980,75:287-292. [6] Reich S.Extension problems for accretive sets in Banach spaces[J].J Funct Anal,1997,26:378-395. [7] Chang S S,Cho Y J,Jung J S,et al.Iterative approximation of fixed point and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces[J].J Math Anal Appl,1998,224:149-165. [8] Xu H K.Existence and convergence for fixed points of mappings of asymptotically nonexpansive type[J].Nonlinear Anal TMA,1991,16(12):1139-1146. [9] Liu Q H.Convergence theorems of the sequence of iterates for asymptotically demi-contractive and hemi-contractive mapings[J].Nonlinear AnalTMA,1996,26(11):1835-1842. [10] Schu J.Iterative construction of fixed points of asymptotically nonexpansive mappings[J].J Math Anal Appl,1991,158:407-413. [11] ZENG Lu-chuan.A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process[J].J Math Anal Appl,1998,226:245-250. [12] Kirk W A,Torrejon R.An asymptotically nonexpansive semigroup in Banach space[J].Nonlinear AnalTMA,1979,1:111-121. [13] Chang S S.Some problems and results in the study of nonlinear analysis[J].Nonlinear Anal TMA,1997,30:4197-4208. -
计量
- 文章访问数: 2161
- HTML全文浏览量: 114
- PDF下载量: 678
- 被引次数: 0
下载:
渝公网安备50010802005915号