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.
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 
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     author = {Kumam, Poom},
     title = {Fixed point theorems for nonexpansive mappings in modular spaces},
     journal = {Archivum mathematicum},
     pages = {345--353},
     publisher = {mathdoc},
     volume = {40},
     number = {4},
     year = {2004},
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     zbl = {1117.47045},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2004__40_4_a2/}
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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/