Let $\rho$ be a modular function satisfying a 

$\Delta_2$-type condition and $L_\rho$ 

the corresponding modular space. The main

result in this paper states that 

if $C$ is a $\rho$-bounded and

$\rho$-a.e sequentially compact subset of 

$L_\rho$ and $ T: C\rightarrow C$ 

is an asymptotically regular mapping such that

$\displaystyle \liminf_{n \rightarrow \infty}|T^n| < 2 $, 

where $|S|$ denotes the Lipschitz constant of $S$, 

then $T$ has a fixed point.  

We show that the estimate $\displaystyle \liminf_{n

\rightarrow \infty}|T^n| < 2 $ cannot be, in general, 

improved.