Geodesic
Fixed point theorems for nonexpansive mappings in modular spaces
Archivum mathematicum, Tome 40 (2004) no. 4, pp. 345-353

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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 $\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.
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.
Classification : 46A80, 46B20, 46E30, 47H09, 47H10
Keywords: fixed point; modular spaces; $\rho $-nonexpansive mapping; $\rho $-normal structure; $\rho $-uniform normal structure; $\rho _r$-uniformly convex 
Kumam, Poom. Fixed point theorems for nonexpansive mappings in modular spaces. Archivum mathematicum, Tome 40 (2004) no. 4, pp. 345-353. http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a2/
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     url = {http://geodesic.mathdoc.fr/item/ARM_2004_40_4_a2/}
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