Let C be a nonempty weakly compact convex subset of a Banach space X,and T:C→C a mapping of asymptotically nonexpansive type.Then there hold the following conclusions:(i)if X has uniform normal structure and lim sup ...Let C be a nonempty weakly compact convex subset of a Banach space X,and T:C→C a mapping of asymptotically nonexpansive type.Then there hold the following conclusions:(i)if X has uniform normal structure and lim sup j→∞|||T^jN|||<√N(X),where |||T^j(N)||| is the exact Lipschitz constant of T^jN,N is some positive integer,and N(X)is the normal structure coefficient of X,then T has a fixed point;(ii) if X is uniformly convex in every direction and has weak uniform normal structure,then T has a fixed point.展开更多
The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive m...The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.展开更多
基金This research is supported both by the Teaching Research Award Fund tor Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C., by the Dawn Program Fund in Shanghai.
文摘Let C be a nonempty weakly compact convex subset of a Banach space X,and T:C→C a mapping of asymptotically nonexpansive type.Then there hold the following conclusions:(i)if X has uniform normal structure and lim sup j→∞|||T^jN|||<√N(X),where |||T^j(N)||| is the exact Lipschitz constant of T^jN,N is some positive integer,and N(X)is the normal structure coefficient of X,then T has a fixed point;(ii) if X is uniformly convex in every direction and has weak uniform normal structure,then T has a fixed point.
文摘The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.
