Fuzzy Fixed Point Results in Convex C ∗ -Algebra-Valued Metric Spaces

Abstract

The purpose of this note is to come up with some new directions in fuzzy fixed point theory. To this effect, notions of a C ∗ -algebra-valued fuzzy λ -contraction and related concepts in a convex C ∗ -algebra-valued metric space ( C ∗ -AVMS) are set-up. In line with the view of a Hausdorff distance function, an idea of a distance between two approximate quantities is proposed. Consequently, two fixed point results of a C ∗ -algebra-valued fuzzy mapping ( C ∗ -AVFM) for the new type of contractions are established using Mann and Ishikawa iterative schemes. For some future investigations of our results, two open problems are noted concerning sufficient criteria guaranteeing the existence of fixed points of a C ∗ -algebra-valued fuzzy λ -contraction and whether or not the Picard iteration for a C ∗ -algebra-valued fuzzy λ -contraction converges.