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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 Tx n =lim n Ix n ((E.A) property); (b) for all x,yX, the inequality

F(d(Tx,Ty),d(Ix,Iy),d(Ix,Tx),d(Iy,Ty),d(Ix,Ty),d(Iy,Tx))0,

where F(t 1 ,t 2 ,t 3 ,t 4 ,t 5 ,t 6 ) is a semi-continuous function + 6 with the following properties: F is non-increasing in the variable t 5 and t 6 , there exists h(0,1) such that, for every u,v0, the relations F(u,v,v,u+v,0)0 and F(u,v,u,v,0,u+v)0 imply uhv, 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.


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