Throughout this paper, we
assume that K is strongly normal, that P = {d(x,y);x,y ∈ X}, that P denotes the
weak closure of P, and that P1= {z;z ∈P and z≠𝒪}. The main result of this paper
is the following.
Let (X,d) be a nonempty K-complete metric space, and let S, T be mappings of
X into itself satisfying (1) and (2).
ϕ(d(Sx,Ty)) ≦ d(x,y), x≠y ∈ X,
(1)
ϕ(t) > t for any t ∈ P1,
(2)
where ϕ : P1→ K is lower semicontinuous on P1.
Then exactly one of the following three statements holds:
S and T have a common fixed point, which is the only periodic point for
both S and T;
There exist a point x0∈ X and an integer p > 1 such that Sx0= x0=Tpx0 and Tx0≠x0;
There exist a point y0∈ X and an integer q > 1 such that Sqy0= y0=Ty0 and Sy0≠y0.
