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An implicit function implies several contraction conditions. (English) Zbl 1180.54052

The article deals with new generalization of coincidence point theorem. It is assumed that A,B,C,D are self-mappings of a metric space (X,d) satisfying, for all x,yX, the inequality

F(d(Ax,By),d(Sx,Ty),d(Ax,Sx),d(By,Ty),d(Sx,By),d(Ty,Ax))0,

where F: + 6 is a function with properties: (F 1 ) F(t,0,t,0,0,t)>0 for all t>0; (F 2 ) F(t,0,0,t,t,0)>0 for all t>0; (F 3 ) F(t,t,0,0,t,t)>0 for all t>0. Furthermore, it is assumed that there exist two sequences {x n },{y n } such that

lim n Ax n =lim n Sx n =lim n By n =lim n Ty n =tforsometX,

and that S(X) and T(X) are closed. Under these conditions, it is stated that both pairs (A,S) and (B,T) have a coincidence point and, moreover, A,B,C,T have a unique common fixed point, provided that both pairs (A,S) and (B,T) are weakly compatible ((A,S) is called weakly compatible if Ax=Sx for some X implies ASx=SAX, and, similarly, for (B,T)). Some modifications of this statement are also given. A number of examples of functions F are presented. Comparisons between old and new results are gathered at the end of the article.


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