# zbMATH — the first resource for mathematics

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 $$x\in X$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
##### Keywords:
bimetric space; $$b$$-metric space
Full Text:
##### References:
 [1] Czerwik S.: Fixed point theorems and special solutions of functional equations. Prace Nauk Univ. Slask. 428 (1980), 1-83. [2] Gantmacher F. R.: Theory of matrix. Russian, Moscow, 1966. [3] Granas A., Dugundi J.: Fixed point theory. Moscow, 1982. [4] Matkowski J.: Integrable solutions of functional equations. Disserationes Math. 127 (1975), 5-63. · Zbl 0318.39005 [5] Zamfirescu T.: Fixpoint theorems in metric spaces. Arch, Math (Basel) 23 (1972), 292-298. · Zbl 0239.54030 · doi:10.1007/BF01304884
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.