Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Opial's modulus and fixed points of semigroups of mappings

Author(s): Tadeusz Kuczumow
Journal: Proc. Amer. Math. Soc. 127 (1999), 2671-2678.
MSC (1991): Primary 47H10, 46B20
Posted: March 16, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: If $X$ is a Banach space with the non-strict Opial property and $r_{X}\left ( 1\right ) >0$ and $C$ is a nonempty convex weakly compact subset of $X$, then every semigroup $\mathfrak{T}=\left \{ T_{t}:t\in G\right \} $ of asymptotically regular selfmappings of $C$ with $\sigma \left ( \mathfrak{T}\right ) <1+r_{X}\left ( 1\right ) $ has a common fixed point.

References:

[1]
BAILLON J.-B., Quelques aspects de la théorie des points fixes dans les espaces de Banach I, Séminaire d'Analyse Fonctionnelle de l'Ecole Polytechnique VII (1978-1979). MR 81d:47036

[2]
BROWDER F.E. & PETRYSHYN W.V., The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571-576. MR 32:8155b

[3]
M. BUDZY\'{N}SKA, T. KUCZUMOW & S. REICH, A uniformly asymptotic normal structure, a semi-Opial property of Banach spaces and fixed points of uniformly lipschitzian semigroups. Part I, Abstr. Appl. Anal. (to appear).

[4]
M. BUDZY\'{N}SKA, T. KUCZUMOW & S. REICH, A uniformly asymptotic normal structure, a semi-Opial property of Banach spaces and fixed points of uniformly lipschitzian semigroups. Part II, Abstr. and Appl. Anal. (to appear).

[5]
BYNUM W. L., Normal structure coefficients for Banach spaces, Pacif. J. Math. 86 (1980), 427-436. MR 81m:46030

[6]
DOMÍNGUEZ BENAVIDES T., Fixed point theorems for uniformly Lipschitzian mappings and asymptotically regular mappings, Nonlinear Analysis 32 (1998), 15-27. CMP 98:07

[7]
DOMÍNGUEZ BENAVIDES T., Stability of the fixed point property for nonexpansive mappings, preprint.

[8]
DOMÍNGUEZ BENAVIDES T. & XU H.-K., A new geometrical coefficient for Banach spaces and its applications in fixed point theory, Nonlinear Analysis 25 (1995), 311-325. MR 96e:47062

[9]
EDELSTEIN M. & O'BRIEN R.C., Nonexpansive mappings, asymptotic regularity, and successive approximations, J. London Math. Soc. 17 (1978), 547-554. MR 80b:47074

[10]
GOEBEL K. & KIRK W.A., A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135-140. MR 49:1242

[11]
GOEBEL K. & KIRK W.A., Iteration processes for nonexpansive mappings,, Topological Methods in Nonlinear Functional Analysis ( S.P. Singh and S. Thomeier, Eds), Contemporary Mathematics, American Mathematical Society, Providence, RI, vol. 21, 1983, pp. 115-123. MR 85a:47059

[12]
GOEBEL K. & KIRK W.A., Topics in metric fixed point theory, Cambridge University Press, 1990. MR 92c:47070

[13]
GÓRNICKI J., Fixed points of asymptotically regular semigroups in Banach spaces, Rend. Circ. Mat. Palermo 46 (1997), 89-118. MR 98f:47064

[14]
GÓRNICKI J. & KRÜPPEL M., Fixed points of uniformly lipschitzian mappings, Bull. Polish Acad. Sci. Math. 36 (1988), 57-63. MR 91a:47079

[15]
ISHIKAWA S., Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71. MR 54:1030

[16]
JIMÉNES-MELADO A. & LLORENS-FUSTER E., Stability of the fixed point property for nonexpansive mappings, Houston J. of Math. 18 (1992), 251-257.

[17]
JIMÉNES-MELADO A. & LLORENS-FUSTER E., Opial modulus and stability of the fixed point property, preprint.

[18]
KIM T.H. & KIRK W.A., Fixed point theorems for Lipschitzian mappings in Banach spaces, Nonlinear Analysis 26 (1996), 1905-1911. MR 97b:47061

[19]
KRASNOSELSKII M.A., Two observations about the method of successive approximations, Uspehi Mat. Nauk 10 (1955), 123-127. MR 16:833a

[20]
LIM T.-C. & XU H.-K., Uniformly Lipschitzian mappings in metric spaces with uniform normal structure, Nonlinear Analysis 25 (1995), 1231-1235. MR 96i:54039

[21]
LIN P.-K., A Uniformly asymptotically regular mapping without fixed points, Canad. Math. Bull. 30 (1987), 481-483. MR 89a:47080

[22]
LIN P.-K., Stability of the fixed point property of Hilbert spaces, preprint.

[23]
LIN P.-K., TAN K.-K. & XU H.-K., Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Analysis 24 (1995), 929-946. MR 96a:47094

[24]
OPIAL Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. MR 35:2183

[25]
PRUS S., Banach spaces with the uniform Opial property, Nonlinear Analysis 18 (1992), 697-704. MR 93h:46022

[26]
TINGLEY D., Noncontractive uniformly lipschitzian semigroups in Hilbert space, Proc. Amer. Math. Soc. 92 (1984), 355-361. MR 85k:47105

[27]
TINGLEY D., An asymptotically nonexpansive commutative semigroup with no fixed points, Proc. Amer. Math. Soc. 97 (1986), 107-113. MR 87g:47113

[28]
XU. H.-K., Geometrical coefficients of Banach spaces and nonlinear mappings, Recent Advances on metric fixed point theory (T. Domínguez Benavides, ed.), Universidad de Sevilla, Serie: Ciencias, Núm. 48 (1996), 161-178. MR 98f:46012

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H10, 46B20

Retrieve articles in all Journals with MSC (1991): 47H10, 46B20

Additional Information:

Tadeusz Kuczumow
Affiliation: Instytut Matematyki, UMCS, 20-031 Lublin, Poland
Email: tadek@golem.umcs.lublin.pl

DOI: 10.1090/S0002-9939-99-04805-4
PII: S 0002-9939(99)04805-4
Keywords: Opial's modulus, semigroups of mappings, fixed points
Received by editor(s): February 27, 1997
Received by editor(s) in revised form: September 25, 1997 and November 14, 1997
Posted: March 16, 1999
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1999, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google