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) $$d(x,y)\leq d(x,z)+ d(z,y)$$ 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 $d(Tx, Ty)\leq \alpha\{d(x,Tx)+ d(y,Ty)\},$ $$\forall x,y\in X$$ and $$0\leq\alpha<{1\over 2}$$, 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 $d(fx,gy)\leq h\max\{d(x,y), d(x,fx), d(y,gy)\}$ for all $$x,y\in X$$. Then $$f$$ and $$g$$ have a unique common fixed point.

MSC:

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

Keywords:

dislocated quasi-metric; fixed point
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