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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×X[0,) be a function satisfying (i) d(x,y)=d(y,x)=0x=y, (ii) d(x,y)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:XX is a continuous mapping satisfying

d(Tx,Ty)α{d(x,Tx)+d(y,Ty)},

x,yX and 0α<1 2, then T has a unique fixed point.

Other results include

Theorem. Let (X,d) be a complete dislocated quasi-metric space. Let T:XX 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:XX be continuous mappings satisfying

d(fx,gy)hmax{d(x,y),d(x,fx),d(y,gy)}

for all x,yX. Then f and g have a unique common fixed point.


MSC:
54H25Fixed-point and coincidence theorems in topological spaces