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

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.

### MSC:

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

Zbl 0291.54056
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