Jungck's common fixed point theorem and E.A property (Q928238)

This article deals with fixed points of mappings in a metric space \((X,d)\). The main result is concerned with points of coincidence for a pair of self-mappings \(T\) and \(I\). It is assumed that: (a) there exists a sequence \(\{x_n\}\) in \(X\) such that \(\lim_{n\to\infty}Tx_n=\lim_{n\to\infty}Ix_n\) ((E.A)\ property); (b) for all \(x,y \in X\), the inequality \[ F(d(Tx,Ty),d(Ix,Iy),d(Ix,Tx),d(Iy,Ty),d(Ix,Ty),d(Iy,Tx)) \leq 0, \] where \(F(t_1,t_2,t_3,t_4,t_5,t_6)\) is a semi-continuous function \({\mathbb R}^6_+ \to {\mathbb R}\) with the following properties: \(F\) is non-increasing in the variable \(t_5\) and \(t_6\), there exists \(h\in (0,1)\) such that, for every \(u,v\geq 0\), the relations \(F(u,v,v,u+v,0)\leq 0\) and \(F(u,v,u,v,0,u+v)\leq 0\) imply \(u\leq hv\), and \(F(u,u,0,0,u,u)>0\) for all \(u>0\) (the authors call such functions ``implicit functions of Popa''); and (c) \(I(X)\) is a complete subspace of \(X\). Under these assumptions, the pair \((T,I)\) has a point of coincidence. Moreover, under the additional assumption that \((T,I)\) is weakly compatible, the pair \((T,I)\) has a common fixed point. As application, the problem of the existence of common fixed points for two finite families of mappings is considered. The article also presents some illustrative examples.