Common fixed points of two mappings

Chi Song Wong

Vol. 48 (1973), No. 1, 299–312

DOI: 10.2140/pjm.1973.48.299

Abstract

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,α_s,α₄,α₅ of (0,∞) into [0,1) such that (a) Σ_i=1⁶α_i < 1; (b) α₁ = α₂ or α₈ = α₄, (c) lim_t↓0(α₁ + α₂) < 1 and lim_t↓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 a_i = α_i(d(x,y))