Abstract. In this note, we prove some fixed point theorems for Kannan type mappings. We will use the additional conditions as compactness or asymptotic regularity or involutions. Our proofs are inspired by the study of Lipschitzian mappings (Agarwal et al., Fixed point theory for Lipschitzian-type mappings with applications, 2009).Mathematics Subject Classification. Primary 47H10; Secondary 54H25.

Recently, Wardowski (Fixed Point Theory Appl. 2012:94, 2012 introduced a new concept of F-contraction and proved a fixed point theorem which generalizes the Banach contraction principle. Following this direction of research, in this paper, we present some new fixed point results for F-expanding mappings, especially on a complete G-metric space. MSC: Primary 47H10; secondary 54H25

Asymptotic regularity allows to provide simple proofs of Banach's theorem and Kannan's theorem. Using asymptotic regularity and Kannan's type conditions we generalize these results, in particular, the Banach contraction principle (see Theorem 2.6 and Corollary 2.10). Further, we discuss the analogous results for monotone mappings on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Next, we will prove a random version of the presented deterministic fixed-point theorems.

We present an analog of Banach's fixed point theorem for CJM contractions in preordered metric spaces. These results substantially extend theorems of Ran and Reurings' (Proc. Am. Math. Soc. 132(5): 1435-1443, 2003 and Nieto and Rodríguez-López (Order 22: 223-239, 2005). MSC: Primary 47H10; secondary 54H25Keywords: contractive map; fixed point; preordered space; G-metric space DefinitionsWe have to introduce many types of contractions.Definition . Let T be a mapping on a metric space (X, d).. T is said to be a (usual) contraction (C, for short) [], if there exists λ ∈ [, ) such that d(Tx, Ty) ≤ λd(x, y) for any x, y ∈ X.. T is said to be a Browder contraction (Bro, for short) [], if there exists a function ϕ from (, ∞) into itself satisfying the following:

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