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The results on fixed points in dislocated and dislocated quasi-metric space. (English) Zbl 1216.54009
Let $X$ be a nonempty set and let $d: X\times X\to [0,\infty)$ be a function satisfying (i) $d(x,y)= d(y,x)= 0\Rightarrow x= y$, (ii) $d(x,y)\le d(x,z)+ d(z,y)$ for all $x,y,z$ in $X$. Then $d$ is called a dislocated quasi-metric for $X$. The couple $(X,d)$ is known as a dislocated quasi-metric space. A typical result in this paper is given below. Theorem. Let $(X,d)$ be a complete dislocated quasi-metric space. If $T: X\to X$ is a continuous mapping satisfying $$d(Tx, Ty)\le \alpha\{d(x,Tx)+ d(y,Ty)\},$$ $\forall x,y\in X$ and $0\le\alpha<{1\over 2}$, then $T$ has a unique fixed point. Other results include Theorem. Let $(X,d)$ be a complete dislocated quasi-metric space. Let $T: X\to X$ be a continuous generalized contraction. Then $T$ has a unique fixed point. Theorem. Let $(X,d)$ be a complete dislocated metric space. Let $f,g: X\to X$ be continuous mappings satisfying $$d(fx,gy)\le h\max\{d(x,y), d(x,fx), d(y,gy)\}$$ for all $x,y\in X$. Then $f$ and $g$ have a unique common fixed point.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces
##### Keywords:
dislocated quasi-metric; fixed point
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