Asymptotic behavior of nonexpansive sequences

and mean points

Authors:
Jong Soo Jung and Jong Seo Park

Journal:
Proc. Amer. Math. Soc. **124** (1996), 475-480

MSC (1991):
Primary 47H09

MathSciNet review:
1291776

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real Banach space with norm and let be a nonexpansive sequence in (i.e., for all ). Let . We deal with the mean point of concerning a Banach limit. We show that if is reflexive and , then and there exists a unique point with such that . This result is applied to obtain the weak and strong convergence of .

**1.**Viorel Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843****2.**Behzad Djafari Rouhani,*Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 951–956. MR**1120510**, 10.1090/S0002-9939-1993-1120510-8**3.**Ky Fan and Irving Glicksberg,*Some geometric properties of the spheres in a normed linear space*, Duke Math. J.**25**(1958), 553–568. MR**0098976****4.**Kazimierz Goebel and Simeon Reich,*Uniform convexity, hyperbolic geometry, and nonexpansive mappings*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR**744194****5.**Elon Kohlberg and Abraham Neyman,*Asymptotic behavior of nonexpansive mappings in uniformly convex Banach spaces*, Amer. Math. Monthly**88**(1981), no. 9, 698–700. MR**643273**, 10.2307/2320677**6.**Elon Kohlberg and Abraham Neyman,*Asymptotic behavior of nonexpansive mappings in normed linear spaces*, Israel J. Math.**38**(1981), no. 4, 269–275. MR**617673**, 10.1007/BF02762772**7.**A. Pazy,*Asymptotic behavior of contractions in Hilbert space*, Israel J. Math.**9**(1971), 235–240. MR**0282276****8.**Andrew T. Plant and Simeon Reich,*The asymptotics of nonexpansive iterations*, J. Funct. Anal.**54**(1983), no. 3, 308–319. MR**724526**, 10.1016/0022-1236(83)90003-4**9.**Simeon Reich,*On the asymptotic behavior of nonlinear semigroups and the range of accretive operators*, J. Math. Anal. Appl.**79**(1981), no. 1, 113–126. MR**603380**, 10.1016/0022-247X(81)90013-5**10.**Simeon Reich,*On the asymptotic behavior of nonlinear semigroups and the range of accretive operators. II*, J. Math. Anal. Appl.**87**(1982), no. 1, 134–146. MR**653610**, 10.1016/0022-247X(82)90157-3**11.**Wataru Takahashi,*The asymptotic behavior of nonlinear semigroups and invariant means*, J. Math. Anal. Appl.**109**(1985), no. 1, 130–139. MR**796047**, 10.1016/0022-247X(85)90181-7

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Additional Information

**Jong Soo Jung**

Affiliation:
Department of Mathematics, Dong–A University, Pusan 604–714, Korea

Email:
jungjs@seanghak.donga.ac.kr.

**Jong Seo Park**

Affiliation:
Department of Mathematics, Graduate School, Dong-A University, Pusan 604–714, Korea

DOI:
https://doi.org/10.1090/S0002-9939-96-03039-0

Keywords:
Asymptotic behavior,
Banach limit,
mean point,
nonexpansive\linebreak \ sequence

Received by editor(s):
March 24, 1994

Received by editor(s) in revised form:
August 22, 1994

Additional Notes:
This research was supported by the Korea Science and Engineering Foundation, project number 941-0100-035-2.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
American Mathematical Society