## Fixed point theorems by altering distances between the points.(English)Zbl 0553.54023

Main result. Let (X,d) be a complete metric space, $$\phi$$ : $${\mathbb{R}}_+\to {\mathbb{R}}_+$$ an increasing continuous function such that $$\phi (t)=0$$ if and only if $$t=0$$. Let a, b, c be three decreasing functions from $${\mathbb{R}}_+\setminus \{0\}$$ into [0,1) such that $$a(t)+2b(t)+c(t)<1$$ for every $$t>0$$. Suppose that mapping $$T: X\to X$$ satisfies the following condition: $\phi (d(Tx,Ty))\leq a(d(x,y))\cdot \phi (d(x,y))+b(d(x,y))\cdot$
$\cdot \{\phi (d(x,Tx))+\phi (d(y,Ty))\}+c(d(x,y))\cdot \min \{\phi (d(x,Ty)),\phi (d(y,Tx))\},$ for x,y$$\in X$$, $$x\neq y$$. Then T has a unique fixed point.
Reviewer: V.V.Obuhovskii

### MSC:

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

### Keywords:

complete metric space
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### References:

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