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Fixed point theorems for a sequence of multifunctions. (English) Zbl 0551.54035

It is proved that if \(\{T_ n\}_{n\in N}\) is a sequence of multifunctions of a complete metric space \((x,d)\) into \(CB(N)\) satisfying the inequality \[ H(T_ 1x,T_ ny)^ 2<k \max [d(x,T_ 1x).d(y,T_ ny);d(x,T_ ny),d(y,T_ 1x); \]
\[ d(x,T_ 1x).d(x,T_ ny);d(y,T_ 1x)d(y,T_ ny);d^ 2(x,y)] \] for all x,y in X, where \(0\leq k<\frac{1}{2}\), \(n\geq 2\) and \(H\) is the Hausdorff-Pompeiu metric on \(CB(X)\), then \(\{T_ n\}_{n\in N}\) has common fixedpoints and \(F(T_ 1)=F(T_ n),\) where \(F(T)=\{x\in N/x\in Tx\}.\) An analogous theorem is proved for a metric space \(X\) with two metrics \(e\) and \(d\). The method used is a combination of methods used by J. Achari [C.R. Acad. Bulg. Sci. 30, 171-174 (1977; Zbl 0371.54075)] and T. Kita [Math. Jap. 22, 113-116 (1977; Zbl 0365.54024)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
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