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Coincidence points for set-valued mappings in convex metric spaces. (English) Zbl 0639.54037

One of the first fixed point theorems - that a contraction mapping has a fixed point - was generalized by N. A. Assad and W. A. Kirk in Pac. J. Math. 43, 553-562 (1972; Zbl 0239.54032) to the case of multivalued mappings f of convex metric spaces, that are contraction mappings in the sense of Hausdorff metric H. The authors generalize this theorem to a coincidence point theorem. Namely, if \(f: K\to M\) is multivalued, S,T: \(K\to M\) single-valued and f is a contraction in the sense that H(fx,fy)\(\leq qd(Sx,Ty)\) for \(q<1\), then under some natural conditions there exists a point z for which Tz\(\in fz\) and Sz\(\in Fz\) (If \(K=M\) and \(S=T=id\) then this is exactly Assad-Kirk theorem). Several likewise results are also proved.
Reviewer: O.M.Kosheleva

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0239.54032
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