Fixed points for four mappings. (English) Zbl 0767.54037

Kang, S. M.; Rhoades, B. E.

Math. Jap. 37, No. 6, 1053-1059 (1992).

Let \(A\), \(B\), \(S\), \(T\) be selfmappings of a complete metric space \((X,d)\) such that at least one of them is continuous; \(A\), \(B\) are surjective; the pairs \(A\), \(S\) and \(B\), \(T\) are compatible in the sense of Jungck; and \(\phi(d(Ax,By))\geq d(Sx,Ty)\), \(x,y\in X\), where \(\phi: [0,\infty)\to [0,\infty)\) is non-decreasing, upper-semicontinuous and \(\phi(t)<t\) \((t>0)\). Under these assumptions the authors prove that \(A\), \(B\), \(T\), \(S\) have a unique common fixed point. This is a generalization of an earlier result of the second author. Some examples show that surjectivity and compatibility conditions are necessary.

Reviewer: J.Matkowski (Bielsko-Biała)

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

(weakly) commuting mappings; compatible mappings; unique common fixed point