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Remarks on some fixed point theorems satisfying implicit relations. (English) Zbl 1033.54025

Let I be the set of all continuous functions F:( + ) 6 satisfying the following conditions: (i) F is non-increasing in t 5 and t 6 ; (ii) there exists h(0,1) such that if u,v0 each of the relations F(u,v,v,u,u+v,0)0 or F(u,v,u,v,0,u+v)0 imply uhv; (iii) F(u,u,0,0,v,v)>0 for all u>0.

The following theorem is the main result of the paper. Let (X,d) be a metric space and S,T,I,J:XX four mappings satisfying the following conditions: (a) S(X)J(X), T(X)I(X) and one of the sets S(X),T(X),I(X) and J(X) is complete; (b) for all x,yX and FJ

F(d(Sx,Ty),d(Ix,Jy),d(Ix,Sx),d(Jy,Ty),d(Ix,Ty),d(Jy,Sx))0·

Then each of the pairs of mappings (S,I) and (T,J) has a coincidence point. Moreover, if each of the pairs of mappings (S,I) and (T,J) commute at their coincidence points, then S,T,I and J have a unique common fixed point.

The previous theorem improves under many aspects an earlier result of V. Popa [Demonstr. Math. 32, 157–163 (1999; Zbl 0926.54030]. Next the author establishes related results and give illustrative examples which demonstrate the utility of the proved results.


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