Asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a Hilbert space

摘要

We extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175) for a commutative semigroup, to a general semitopological semigroup, study the asymptotic behavior of such curves in a Hilbert space, and apply our results to study the asymptotic behavior for almost-orbits of semigroups of asymptotically nonexpansive-type mappings over a general semitopological semigroup, in a Hilbert space. Since in this case the semigroup is not directed, the proofs in Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175) do not extend to this case, and new methods have to be used for the proofs. In addition to Rouhani (J. Math. Anal. Appl. 147 (1990) 465; J. Math. Anal. Appl. 151 (1990) 226; Nonlinear Anal. 44 (2001) 627; Nonlinear Anal. 51 (2002) 735) and Rouhani and Kim (J. Nonlinear Convex Anal. 4 (2003) 175), our results extend and unify many previously known results (J. Math. Anal. Appl. 127 (1987) 206; Proc. Amer. Math. Soc. 104 (1988) 431; Nonlinear Anal. 29 (1997) 539; Topol. Methods Nonlinear Anal. 5 (1995) 305; J. Math. Anal. Appl. 105 (1985) 514; Nonlinear Anal. 26 (1996) 1411; Pacific J. Math. 126 (1987) 277; J. Math. Anal. Appl. 206 (1997) 451; Chinese Ann. Math. 6 (1996) 729; Nonlinear Anal. 14 (1990) 69; J. Math. Anal. Appl. 85 (1982) 172; Proc. Amer. Math. Soc. 81 (1981) 253; Proc. Amer. Math. Soc. 97 (1986) 55; Canad. J. Math. 44 (1992) 880; J. Math. Anal. Appl. 142 (1989) 242; Proc. Amer. Math. Soc. 117 (1993) 385). (C) 2004 Elsevier Ltd. All rights reserved.