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Coincidence points of compatible multivalued mappings. (English) Zbl 0890.54045

Let \((X,d)\) be a metric space and \(CB(X)\) the set of all nonempty closed bounded subsets of \((X,d)\). In [O. Hadžić and L. Gajić, Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 16, No. 1, 13-25 (1986; Zbl 0639.54037)] Hadžić introduced the following definition: The mappings \(A:X\to CB(X)\) and \(S:X\to X\) are said to be weakly commuting iff for every \(x\in X\) and \(y\in X\) such that \(y\in Ax\) we have \(d(Sy,SAx)\leq d(Sx,Ax)\). The authors extend the concept of compatibility for single-valued mappings [G. Jungck, Int. J. Math. Math. Sci. 9, 771-779 (1986; Zbl 0613.54029)] to the setting of single-valued and multivalued mappings in the following way: The mapping \(A:X\to CB(X)\) and \(S:X\to X\) are said to be compatible if \(\lim d(Sy_n, ASx_n)=0\) whenever \(\{x_n\}\) and \(\{y_n\}\) are sequences in \(X\) such that \(\lim Sx_n=\lim y_n=z\) for some \(z\) in \(X\) where \(y_n\in Ax_n\) for \(n=1,2,\dots\). Every weakly commuting pair of mappings \((A,S)\) is compatible but the converse is not true. A point \(z\) in \(X\) is said to be a coincidence point of \(S\) and \(A\) if \(Sz\in Az\). In this paper the authors prove a coincidence theorem replacing in Theorem 1 from [A. Constantin, Math. Jap. 36, No. 5, 925-933 (1991; Zbl 0757.54016)] weakly commuting mappings with compatible mappings generalizing also results from [O. Hadžić, Stud. Univ. Babeş-Bolyai, Math. 26, No. 4, 65-67 (1981; Zbl 0506.54041); T. Kubiak, ibid. 30, 65-68 (1985; Zbl 0579.54037); V. Popa, Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 18, No. 1, 149-156 (1988; Zbl 0714.54041)].
Reviewer: V.Popa (Bacau)

MSC:

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