an:00898043 Zbl 0849.54036 Czerwik, S. Contraction mappings in $$b$$-metric spaces EN Acta Math. Inform. Univ. Ostrav. 1, 5-11 (1993). 00034952 1993
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54H25 47H10 bimetric space; $$b$$-metric space Introducing the concept of a bimetric space a few fixed point theorems have been proved. Let $$X$$ be a nonempty set and $$\mathbb{R}_+$$ the set of all nonnegative numbers. Then $$d: X\times X\to \mathbb{R}_+$$ is a $$b$$-metric iff for all $$x, y, z\in X$$ and all $$r>0$$ the following conditions are satisfied: $$d(x, y)=0 \iff x=y$$; $$d(x, y)= d(y, x)$$; $$d(x, y)< r$$ and $$d(x, z)< r$$ imply $$d(y, z)< 2r$$. The pair $$(X, d)$$ is a $$b$$-metric space. The following result is included as a sample. Let $$(X, d)$$ be a complete $$b$$-metric space and $$T: X\to X$$ satisfy $$d(T (x), T(y))\leq Q(d(x,y))$$, $$x,y\in X$$, where $$Q: \mathbb{R}_+\to \mathbb{R}_+$$ is an increasing function such that $$\lim_{n\to \infty} Q^n (t)=0$$ for each fixed $$t>0$$. then $$T$$ has a unique fixed point $$u$$ and $$\lim_{n\to \alpha} d(T^n (x), u)=0$$ for each $$x\in X$$. S.P.Singh (St.John's)