On Kannan fixed point principle in generalized metric spaces. (English) Zbl 1171.54032

Miheţ, Dorel

J. Nonlinear Sci. Appl. 2, No. 2, 92-96 (2009).

A pair \((X, d)\) is said to be a generalized metric space (in the sense of Branciari) if \(X\) is a nonempty set and \(d:X\times X\rightarrow \mathbb{R}_+ \) is a function which satisfies, for all \(x, y\in X\), the following conditions: (i) \(d(x,y)=0\) iff \(x=y\); (ii) \(d(x,y)=d(y,x)\); (iii) \(d(x,y)\leq d(x,z) + d(z,w)+ d(w,y) \), for each \(z,w\in X\) with \(x\neq z \neq w \neq x\). If \(T\) is a self mapping of a generalized metric space \((X,d)\), we say that \(X\) is \(T\)-orbitally complete if every Cauchy sequence, contained in the orbit of a point under \(T\), converges in \(X\).
In this paper the author gives a new proof, based on L. B. Ćirić’s generalization of the Banach principle [Proc. Am. Math. Soc. 45, 267–273 (1974; Zbl 0291.54056)] for the following result, due to P. Das:
Let \((X,d)\) be a generalized metric space and \(T:X\rightarrow X\) be a mapping such that for some \(\beta \in (0, \frac{1}{2})\) and for all \(x,y\in X\), \(d(Tx, Ty)\leq \beta \big[ d(x, Tx)+ d(y,Ty)\big]\). If \(X\) is \(T\)-orbitally complete, then \(T\) has a unique fixed point.

Reviewer: Mircea Balaj (Oradea)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

generalized metric spaces; \(T\)-orbitally complete space; fixed point

Citations:

Zbl 0291.54056