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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) \(d(x,y)\leq d(x,z)+ d(z,y)\) 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)
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