Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces

Authors:
Kok-Keong Tan and Hong Kun Xu

Journal:
Proc. Amer. Math. Soc. **117** (1993), 385-393

MSC:
Primary 47H20; Secondary 47A35, 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1993-1111223-7

MathSciNet review:
1111223

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Abstract: Let be a nonempty closed convex subset of a Hilbert space , be a continuous nonlinear asymptotically nonexpansive semigroup acting on with a nonempty fixed point set , and be an almost-orbit of . Then almost converges weakly to a fixed point of , i.e., there exists an element in such that

**[1]**J. B. Baillon and R. Schöneberg,*Asymptotic normal structure and fixed points of nonexpansive mappings*, Proc. Amer. Math. Soc.**81**(1981), 257-264. MR**593469 (82c:47068)****[2]**R. E. Brack,*Asymptotic behavior of nonexpansive mappings*, Contemp. Math., vol. 18, Amer. Math. Soc., Providence, RI, 1983, pp. 1-47, MR**728592 (85d:47057)****[3]**M. Edelstein,*The construction of an asymptotic center with a fixed-point property*, Bull. Amer. Math. Soc.**78**(1972), 206-208. MR**0291917 (45:1005)****[4]**-,*Fixed point theorems in uniformly convex Banach spaces*, Proc. Amer. Math. Soc.**44**(1974), 369-374. MR**0358451 (50:10917)****[5]**W. A. Kirk,*Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type*, Israel J. Math.**17**(1974), 339-346. MR**0346605 (49:11330)****[6]**W. A. Kirk and R. Torrejón,*Asymptotically nonexpansive semigroups in Banach spaces*, Nonlinear Anal.**3**(1979), 111-121. MR**520478 (82a:47062)****[7]**A. T. Lau,*Semigroup of nonexpansive mappings on Hilbert space*, J. Math. Anal. Appl.**105**(1985), 514-522. MR**778484 (86m:47085)****[8]**T C. Lim,*On asymptotic centers and fixed points of nonexpansive mappings*, Canad. J. Math.**32**(1980), 421-430. MR**571935 (81k:47081)****[9]**I. Miyadera and K. Kobayasi,*On the asymptotic behavior of almost-orbits of nonlinear contraction semigroups in Banach spaces*, Nonlinear Anal.**6**(1982), 349-365. MR**654811 (84c:47064)****[10]**W. Takahashi and P.-J. Zhang,*Asymptotic behavior of almost-orbits of reversible semigroups of lipschitzian mappings*, J. Math. Anal. Appl.**142**(1989), 242-249. MR**1011422 (90g:47121)****[11]**Z. Y. You and H. K. Xu,*An ergodic convergence theorem for mappings of asymptotically nonexpansive type*, Chinese Ann. Math. Ser. A**11**(1990), 121-125. (Chinese) MR**1074464 (91k:47137)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1111223-7

Keywords:
Almost-orbit,
asymptotically nonexpansive semigroup,
weakly asymptotically regular,
nonexpansive retraction,
fixed point,
metric projection,
asymptotic center

Article copyright:
© Copyright 1993
American Mathematical Society