Let be a nonempty set and let be a function satisfying (i) , (ii) for all in . Then is called a dislocated quasi-metric for . The couple is known as a dislocated quasi-metric space. A typical result in this paper is given below.
Theorem. Let be a complete dislocated quasi-metric space. If is a continuous mapping satisfying
and , then has a unique fixed point.
Other results include
Theorem. Let be a complete dislocated quasi-metric space. Let be a continuous generalized contraction. Then has a unique fixed point.
Theorem. Let be a complete dislocated metric space. Let be continuous mappings satisfying
for all . Then and have a unique common fixed point.