Fixed point theorems for Kannan-type maps

for all x, y ∈ X. Kannan [] proved that if X is complete, then a Kannan mapping has a fixed point. It is interesting that Kannan’s theorem is independent of the Banach contraction principle []. Also, Kannan’s fixed point theorem is very important because Subrahmanyam [] proved that Kannan’s theorem characterizes the metric completeness. That is, a metric space X is complete if and only if every Kannan mapping on X has a fixed point. Using the concept of Hausdorff metric, Nadler [] proved the fixed point theorem for multi-valued contraction maps, which is a generalization of the Banach contraction principle []. Since then various fixed point results concerningmulti-valued contractions have appeared; for example, see [–] and the references cited there. Without using the concept of Hausdorff metric, most recently Dehaish and Latif [] generalized fixed point theorems of Latif andAbdou [], Suzuki [], Suzuki andTakahashi []. In , Kada et al. [] introduced the notion of w-distance and improved several classical results including Caristi’s fixed point theorem. Suzuki and Takahashi [] introduced single-valued and multi-valued weakly contractive maps with respect to w-distance and proved fixed point results for such maps. Generalizing the concept of w-distance, in , Suzuki [] introduced the notion of τ -distance on a metric space and improved several classical results including the corresponding results of Suzuki and Takahashi []. In , Ume [] introduced the new concept of a distance called u-distance, which generalizes w-distance, Tataru’s distance and τ -distance. Then he proved a new minimization theorem and a new fixed point theorem by using u-distance on a complete metric space. Distances in uniform spaces were given byVályi [].More general concepts of distances were given by Wlodarczyk and Plebaniak [–] and Wlodarczyk []. In this paper, we introduce the new classes of Kannan-type multi-valued maps without using the concept of Hausdorff metric and Kannan-type single-valued maps with respect