Common fixed points under Lipschitz type condition. (English) Zbl 1155.54027

Pant, Vyomesh

Bull. Korean Math. Soc. 45, No. 3, 467-475 (2008).

This paper contains five theorems. These results include the following: Let \(f\) and \(g\) be noncompatible pointwise \(R\)-weakly commuting self-mappings of a metric space \((X,d)\) satisfying
(i) \(\overline{fX}\subset gX\), where \(\overline{fX}\) denotes the closure of range of \(f\),
(ii) \(d(fx,fy)\leq k\), \(d(gx,gy)\), \(k\geq 0\), and
(iii) \(d(fx,f^2x)< \max\{d(gx, gfz)\), \(d(g^2x, gfx)\), \(d(fx,gx)\), \(d(f^2x, gfx)\), \(d(fx,gfx)\), \(d(gx,f^2x)\}\),
whenever \(fx\neq f^2x\). Then \(f\) and \(g\) have a common fixed point.

Reviewer: K. Chandrasekhara Rao (Kumbakonam)

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

Lipschitz type mapping pairs; contractive conditions; property (E.A.) noncompatible mappings