Fixed points of asymptotically regular mappings in spaces with uniformly normal structure

Jarosław Górnicki

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 639-643
  • ISSN: 0010-2628

Abstract

top
It is proved that: for every Banach space X which has uniformly normal structure there exists a k > 1 with the property: if A is a nonempty bounded closed convex subset of X and T : A → A is an asymptotically regular mapping such that lim inf n → ∞ | | | T n | | | < k , where | | | T | | | is the Lipschitz constant (norm) of T , then T has a fixed point in A .

How to cite

top

Górnicki, Jarosław. "Fixed points of asymptotically regular mappings in spaces with uniformly normal structure." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 639-643. <http://eudml.org/doc/247296>.

@article{Górnicki1991,
abstract = {It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that \[ \liminf \_\{n\rightarrow \infty \} |\hspace\{-0.8pt\}|\hspace\{-0.8pt\}|T^n|\hspace\{-0.8pt\}|\hspace\{-0.8pt\}|< k, \] where $|\hspace\{-0.8pt\}|\hspace\{-0.8pt\}|T|\hspace\{-0.8pt\}|\hspace\{-0.8pt\}|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.},
author = {Górnicki, Jarosław},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {asymptotically regular mappings; uniformly normal structure; fixed points; uniformly normal structure; asymptotically regular mapping; Lipschitz constant; fixed point},
language = {eng},
number = {4},
pages = {639-643},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fixed points of asymptotically regular mappings in spaces with uniformly normal structure},
url = {http://eudml.org/doc/247296},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Górnicki, Jarosław
TI - Fixed points of asymptotically regular mappings in spaces with uniformly normal structure
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 639
EP - 643
AB - It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that \[ \liminf _{n\rightarrow \infty } |\hspace{-0.8pt}|\hspace{-0.8pt}|T^n|\hspace{-0.8pt}|\hspace{-0.8pt}|< k, \] where $|\hspace{-0.8pt}|\hspace{-0.8pt}|T|\hspace{-0.8pt}|\hspace{-0.8pt}|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.
LA - eng
KW - asymptotically regular mappings; uniformly normal structure; fixed points; uniformly normal structure; asymptotically regular mapping; Lipschitz constant; fixed point
UR - http://eudml.org/doc/247296
ER -

References

top
  1. Browder F.E., Petryshyn V.W., The solution by iteration of nonlinear functional equations in Banach spaces, Bull. AMS 72 (1966), 571-576. (1966) Zbl0138.08202MR0190745
  2. Bynum W.L., Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427-436. (1980) Zbl0442.46018MR0590555
  3. Casini E., Maluta E., Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Anal., TMA 9 (1985), 103-108. (1985) Zbl0526.47034MR0776365
  4. Daneš J., On densifying and related mappings and their applications in nonlinear functional analysis, in: Theory of Nonlinear Operators (Proc. Summer School, October 1972, GDR), Akademie-Verlag, Berlin, 1974, 15-56. MR0361946
  5. Downing D.J., Turett B., Some properties of the characteristic convexity relating to fixed point theory, Pacific J. Math. 104 (1983), 343-350. (1983) MR0684294
  6. Edelstein M., O'Brien C.R., Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc. (2) 17 (1978), 547-554. (1978) Zbl0421.47031MR0500642
  7. Gillespie A.A., Williams B.B., Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure, Appl. Anal. 9 (1979), 121-124. (1979) MR0539537
  8. Górnicki J., A fixed point theorem for asymptotically regular mappings, to appear. MR1201441
  9. Krüppel M., Ein Fixpunktsatz für asymptotisch reguläre Operatoren in gleichmäßig konvexen Banach-Räumen, Wiss. Z. Pädagog. Hochsch. ``Liselotte Herrmann'' Güstrow, Math.-naturwiss. Fak. 25 (1987), 241-246. (1987) MR0971250
  10. Lin P.K., A uniformly asymptotically regular mapping without fixed points, Canad. Math. Bull. 30 (1987), 481-483. (1987) Zbl0645.47050MR0919440
  11. Yu X.T., On uniformly normal structure, Kexue Tongbao 33 (1988), 700-702. (1988) Zbl0681.46020
  12. Yu X.T., A geometrically aberrant Banach space with uniformly normal structure, Bull. Austral. Math. Soc. 38 (1988), 99-103. (1988) Zbl0646.46017MR0968233

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.