Fixed point theorems for nonexpansive mappings in modular spaces

Poom Kumam

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


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.