Let . Suppose that a mapping satisfies:
(a) if and only if ,
(b) for all , with .
(c) for all , with ,
(d) (symmetry in all three variables),
(e) for all .
Then is called a -metric on and is called a -metric space.
In the present paper the authors, using the setting of -metric space, prove a fixed point theorem for one map, and several fixed point theorems for two maps. They prove, for example:
Theorem 2.5. Let be a -metric space. Suppose that satisfy one of the following conditions:
for all , where . If the range of contains the range of and is a complete subspace of , then and have a unique point of coincidence in . Moreover, if and are weakly compatible, then and have a unique common fixed point.