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Contraction mappings in \(b\)-metric spaces. (English) Zbl 0849.54036
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 \(x\in X\).

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