Fixed point theorems for nonexpansive mappings in modular spaces

Poom Kumam

Archivum Mathematicum (2004)

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

Abstract

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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 ρ is a convex, ρ -complete modular space satisfying the Fatou property and ρ r -uniformly convex for all r > 0 , C a convex, ρ -closed, ρ -bounded subset of X ρ , T : C → C a ρ -nonexpansive mapping, then T has a fixed point.

How to cite

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Kumam, 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

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  11. Kumam P., Fixed Point Property in Modular Spaces, Master Thesis, Chiang Mai University (2002), Thailand. 
  12. Megginson R. E., An introduction to Banach space theory, Graduate Text in Math. Springer-Verlag, New York 183 (1998). (1998) Zbl0910.46008MR1650235
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