Let S,T be functions on a
nonempty complete metric space (X,d). The main result of this paper is the
following. S or T has a fixed point if there exist decreasing functions α1,α2,αs,α4,α5
of (0,∞) into [0,1) such that (a) Σi=16αi< 1; (b) α1= α2 or α8= α4, (c)
limt↓0(α1+ α2) < 1 and limt↓0(α8+ α4) < 1 and (d) for any distinct x,y in
X,
d(S(x),T(y))
≦ a1d(x,S(x)) + a2d(y,T(y)) + a3d(x,T(y))
+ a4d(y,S(x)) + a5d(x,y),
where ai= αi(d(x,y)). A number of related results are obtained.
