IJPAM: Volume 56, No. 2 (2009)

SOME COMMON FIXED POINT THEOREMS
FOR FUZZY MAPPINGS

Ramendra Krishna Bose
Department of Mathematics
The University of Texas - Pan American
Edinburg, Texas, 78539, USA
e-mail: [email protected]


Abstract.Zadeh [22] introduced the concept of fuzzy sets and Heilpern [11] introduced the concept of fuzzy mappings. Since then both the concepts have been generalized by various researchers. Bose and Sahani [10] and others generalized the results of Heilpern. Many authors considered class of fuzzy sets with nonempty compact (convex) $\alpha-$cut sets in a metric (linear) space, but some have given attention to class of fuzzy sets with nonempty closed and bounded $\alpha-$cut sets in a metric space. Dong Qui and Lan Shu [15] considered $\mathcal{CB}\left(X\right)$ (the class of fuzzy sets with nonempty closed and bounded $\alpha-$cut sets) equipped with the generalized Hausdorff metric. They proved some common fixed point theorems for fuzzy mappings $F_{i}:\mathcal{CB}\left(X\right)\rightarrow\mathcal{CB}\left(X\right)$ satisfying certain inequalities. Dong Qui et al [16] considered the class of fuzzy sets, with nonempty compact $\alpha-$cut sets ( $\mathcal{C}\left(X\right)$) and fuzzy mappings $F_{i}:\mathcal{C}\left(X\right)\rightarrow\mathcal{C}\left(X\right)$. This paper deals with several common fixed point theorems of such fuzzy mappings satisfying various (different) inequalities and these extend the works of various researchers. Following Alber and Guerre-Delabriere [2], we extend the concept of $\phi-$weakly contractive condition to the fuzzy mappings and prove a common fixed point theorem for a pair of such fuzzy mappings. Also we prove a fixed point theorem for a fuzzy mapping which is a $\left(\delta,L\right)-$weak contraction (introduced by Berinde and Berinde [6]).

Received: August 18, 2009

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

Key Words and Phrases: common fixed point, fuzzy mappings, weakly contractive mappings, $\left(\delta,L\right)-$weak contraction, $\Phi-$contraction

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 2