Vol. 101, No. 1, 1982

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Remarks on nonlinear contractions

Kun-Jen Chung

Vol. 101 (1982), No. 1, 41–48
Abstract

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 z𝒪}. 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), xy 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 Tx0x0;
  • There exist a point y0 X and an integer q > 1 such that Sqy0 = y0 = Ty0 and Sy0y0.

Mathematical Subject Classification 2000
Primary: 54H25
Milestones
Received: 3 September 1980
Revised: 18 June 1981
Published: 1 July 1982
Authors
Kun-Jen Chung