The concept of a 2-metric space, introduced by S. GAHLER [l], provides a natural abstraction of the area function for EnoLIDean triangles.
Recently, KHAN and FISHER [2] have introduced a necessary and sufficient condition which guarantees the existence of a common fixed point for a pair of continuous mappings in 2-metric as well as in metric spaces.
Tn this note, we shall prove similar theorems under much weaker conditions. Consequently, we extend fixed point theorems of RHOADES [3], SINGH-!I'IWARI-GUPTA [4] and others. Following GAHLER [l] and WFIITE [6], we give some definitions.
Let X be a nonempty set and d : X x X x X +-[0, 00) a function satisfying the following axioms: (i) d(x, y, z ) = 0 if at least two of z, y, z are equal, (ii) for each two distinct points x , y E X , there exists a point z E X such that 4 x 9 Li, 4 =I= 0, P i ) d(x, Y , 4 = 4 x 9 z, Y ) = d(Y, z, z), exist mappings A , B : X -+ SX 11 TX such that AS = SA, BT = T B sattkjyinq W X , BY) S c max ~( S X , Q), d(Sx, A X ) , d ( T y , By), -[d(Sz, By) + d(Ty, A X ) ] 1 { 2 for all x , y E X where 0 < c < 1. Indeed, A, B, S and 1' then have a common unipue fixed point.