IJPAM: Volume 61, No. 1 (2010)

FIXED POINT THEOREMS FOR MULTI-VALUED
MAPPINGS AND FUZZY MAPPINGS

Ramendra Krishna Bose$^1$, Mrinal Kanti Roychowdhury$^2$
$^{1,2}$Department of Mathematics
The University of Texas - Pan American
1201, West University Drive, Edinburg, TX 78539-2999, USA
$^1$e-mails: rkbose@utpa.edu, rkbose@hotmail.com
$^2$e-mail: roychowdhurymk@utpa.edu


Abstract.In this paper, a common fixed point theorem for a pair of multi-valued mappings which satisfy a generalized $\phi$-weakly contractive condition in a complete metric space setting is proved. A fixed point theorem is also proved when the mapping is a non-self mapping with domain a nonempty closed subset of a complete and convex metric space. A coincidence point theorem concerning a single-valued mapping $f$ and a pair of $\phi$-weakly contractive multi-valued mappings with respect to $f$ is also proved. Next two theorems deal with fuzzy mappings in a metric linear space setting: the first one is a common fixed point theorem for a pair of fuzzy mappings, a fuzzy version of the result obtained for a pair of multi-valued mappings, and the second one is a coincidence point theorem concerning a $\phi$-weakly contractive fuzzy mapping with respect to mapping $f$. These results extend the work of several authors.

Received: March 27, 2010

AMS Subject Classification: 03E72, 47H04, 47H10, 54H25, 54C60

Key Words and Phrases: fixed point, multi-valued mapping, fuzzy mapping, weak contraction

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 1