## Fixed point theorems for nonexpansive mappings in modular spaces

##
*Poom Kumam*

**Address.**

Department of Mathematics,
Faculty of Science, King Mongkut's University of Technology Thonburi
Bangkok 10140, Thailand

**E-mail. **pooom.kum@kmutt.ac.th

**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.

**AMSclassification. **46B20, 46E30, 47H10.

**Keywords. ** Fixed point, modular spaces,
$\rho$-nonexpansive mapping, $\rho$-normal structure,
$\rho$-uniform normal structure, $\rho_r$-uniformly convex.