# Fixed point theorems for nonexpansive mappings in modular spaces

Archivum Mathematicum (2004)

- Volume: 040, Issue: 4, page 345-353
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topKumam, Poom. "Fixed point theorems for nonexpansive mappings in modular spaces." Archivum Mathematicum 040.4 (2004): 345-353. <http://eudml.org/doc/249289>.

@article{Kumam2004,

abstract = {In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if $\rho $ is a convex, $\rho $-complete modular space satisfying the Fatou property and $\rho _r$-uniformly convex for all $r>0$, C a convex, $\rho $-closed, $\rho $-bounded subset of $X_\rho $, $T:C\rightarrow C$ a $\rho $-nonexpansive mapping, then $T$ has a fixed point.},

author = {Kumam, Poom},

journal = {Archivum Mathematicum},

keywords = {fixed point; modular spaces; $\rho $-nonexpansive mapping; $\rho $-normal structure; $\rho $-uniform normal structure; $\rho _r$-uniformly convex; -nonexpansive mapping; -normal structure; -uniform normal structure; -uniformly convex},

language = {eng},

number = {4},

pages = {345-353},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Fixed point theorems for nonexpansive mappings in modular spaces},

url = {http://eudml.org/doc/249289},

volume = {040},

year = {2004},

}

TY - JOUR

AU - Kumam, Poom

TI - Fixed point theorems for nonexpansive mappings in modular spaces

JO - Archivum Mathematicum

PY - 2004

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 040

IS - 4

SP - 345

EP - 353

AB - In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if $\rho $ is a convex, $\rho $-complete modular space satisfying the Fatou property and $\rho _r$-uniformly convex for all $r>0$, C a convex, $\rho $-closed, $\rho $-bounded subset of $X_\rho $, $T:C\rightarrow C$ a $\rho $-nonexpansive mapping, then $T$ has a fixed point.

LA - eng

KW - fixed point; modular spaces; $\rho $-nonexpansive mapping; $\rho $-normal structure; $\rho $-uniform normal structure; $\rho _r$-uniformly convex; -nonexpansive mapping; -normal structure; -uniform normal structure; -uniformly convex

UR - http://eudml.org/doc/249289

ER -

## References

top- Aksoy A. G., Khamsi M. A., Nonstandard methods in fixed point theory, Spinger-Verlag, Heidelberg, New York 1990. (1990) Zbl0713.47050MR1066202
- Ayerbe Toledano J. M., Dominguez Benavides T., and López Acedo G., Measures of noncompactness in metric fixed point theory: Advances and Applications Topics in metric fixed point theory, Birkhäuser-Verlag, Basel, 99 (1997). (1997) MR1483889
- Chen S., Khamsi M. A., Kozlowski W. M., Some geometrical properties and fixed point theorems in Orlicz modular spaces, J. Math. Anal. Appl. 155 No. 2 (1991), 393–412. (1991) MR1097290
- Dominguez Benavides T., Khamsi M. A., Samadi S., Uniformly Lipschitzian mappings in modular function spaces, Nonlinear Analysis 40 No. 2 (2001), 267–278. MR1849794
- Goebel K., Kirk W. A., Topic in metric fixed point theorem, Cambridge University Press, Cambridge 1990. (1990) MR1074005
- Goebel K., Reich S., Uniform convexity, Hyperbolic geometry, and nonexpansive mappings, Monographs textbooks in pure and applied mathematics, New York and Basel, 83 1984. (1984) Zbl0537.46001MR0744194
- Khamsi M. A., Fixed point theory in modular function spacesm, Recent Advances on Metric Fixed Point Theorem, Universidad de Sivilla, Sivilla No. 8 (1996), 31–58. (1996) MR1440218
- Khamsi M. A., Uniform noncompact convexity, fixed point property in modular spaces, Math. Japonica 41 (1) (1994), 1–6. (1994) Zbl0820.47063MR1305537
- Khamsi M. A., A convexity property in modular function spaces, Math. Japonica 44, No. 2 (1990). (1990) MR1416264
- Khamsi M. A., Kozlowski W. M., Reich S., Fixed point property in modular function spaces, Nonlinear Analysis, 14, No. 11 (1990), 935–953. (1990) MR1058415
- Kumam P., Fixed Point Property in Modular Spaces, Master Thesis, Chiang Mai University (2002), Thailand.
- Megginson R. E., An introduction to Banach space theory, Graduate Text in Math. Springer-Verlag, New York 183 (1998). (1998) Zbl0910.46008MR1650235
- Musielak J., Orlicz spaces and Modular spaces, Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York 1034 (1983). (1983) Zbl0557.46020MR0724434
- Musielak J., Orlicz W., On Modular spaces, Studia Math. 18 (1959), 591–597. (1959) Zbl0099.09202MR0101487
- Nakano H., Modular semi-ordered spaces, Tokyo, (1950). (1950)

## NotesEmbed ?

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