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Jungck’s common fixed point theorem and E.A property. (English) Zbl 1158.54021
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 $$F(d(Tx,Ty),d(Ix,Iy),d(Ix,Tx),d(Iy,Ty),d(Ix,Ty),d(Iy,Tx)) \le 0,$$ where $F(t_1,t_2,t_3,t_4,t_5,t_6)$ is a semi-continuous function ${\Bbb R}^6_+ \to {\Bbb R}$ 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 $u,v\ge 0$, the relations $F(u,v,v,u+v,0)\le 0$ and $F(u,v,u,v,0,u+v)\le 0$ imply $u\le hv$, 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.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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