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Common fixed point theorems. (English) Zbl 0767.54038

This paper contains some common fixed point theorems in metric spaces. The main result can be stated as follows: Theorem: Let A, B, S and T be mappings from a complete metric space (X,d) into itself satisfying the conditions: (i) A(X)T(X) and B(X)S(X),

d(Ax,By)h·max{d(Ax,Sx),d(By,Ty),1 2[d(Ax,Ty)+d(By,Sx)],d(Sx,Ty)}( ii )

for all x, y in X, where 0h<1. Further, suppose that (iii) one of A, B, S and T is continuous, (iv) pairs A, S and B, T are compatible on X. Then A, B, S and T have a unique common fixed point in X. The above theorem generalizes several known results due to B. Fisher, G. Jungck, M. S. Khan and M. Imdad. A result for compact metric space is also proved. Examples are given to illustrate all the results.


MSC:
54H25Fixed-point and coincidence theorems in topological spaces